c 2013 by nachiketa chakraborty. all rights reserved
TRANSCRIPT
c© 2013 by Nachiketa Chakraborty. All rights reserved.
STANDARD MODEL FOREGROUNDS IN NUCLEAR AND PARTICLEASTROPHYSICS
BY
NACHIKETA CHAKRABORTY
DISSERTATION
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Astronomy
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2013
Urbana, Illinois
Doctoral Committee:
Professor B. D. Fields, ChairProfessor P. M. RickerAssociate Professor A. J. KemballProfessor J. C. Peng
Abstract
Nuclear and particle astrophysics comprises of applying standard principles of nuclear and
particle physics to astronomical systems and making predictions for observation. Also, ex-
treme environments in astronomical systems often serve as a probe for re-evaluating fun-
damental principles of physics and astronomy. The research described here comprises of
both these aspects of nuclear and particle astrophysics. There are two broad themes : pri-
mordial nucleosynthesis and cosmic gamma-ray sources. As a part of the first theme, the
“cosmological lithium problem”, and a possible nuclear physics solution to it is examined.
The theoretical prediction of the abundance of 7Li produced in the big bang exceeds the
observationally inferred abundance by a factor of 3-4, that is very difficult to explain within
the Standard Model of particle physics and cosmology. The possibility of missed nuclear
resonances in the network of nuclear reactions used to compute the theoretical primordial
abundance is discussed. Three such candidate resonances are found prompting experiments
to either find them or rule them out. In the light of recent experiments eliminating the most
promising of these candidates, requirement of new physics beyond the Standard Model looms
large as a a possible solution. In the second area of research, properties of cosmic gamma-ray
emitting sources such as star forming galaxies and blazars are studied. The gamma-ray sky
has contributions from these guaranteed sources and potentially others such as particle dark
matter. In the light of this, gamma-rays properties of the guaranteed astronomical sources
as spectral slope, photon count statistics and polarisation of X-rays and soft gamma-rays
is studied in the hope of achieving the broad goal of understanding the underlying physics
of radiative emission of these sources and hopefully disentangling these “foregrounds” from
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other exotic sources such as particle dark matter.
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Acknowledgments
My research and life in general throughout my Ph.D. would not have been successful without
the contribution of several people.
First and foremost, my advisor Brian Fields has and continues to be a tremendous in-
stitution for my training as a scientist. In a profession where the input of your mentor is
paramount, I have been fortunate to have a phenomenal scientist and a superlative person
to mould me into a scientist. I have learnt everything from scientific aspects of astrophysics
to the social aspects of building collaborations with other scientists from Brian. Alongside
doing good research, communicating it well is also very important. In this aspect, my com-
munication skills through talks and other means have improved significantly under Brian’s
tutelage. My entire research experience with Brian has been a thoroughly enjoyable expe-
rience. I am very, very lucky to have forged a strong, fruitful collaboration with a scientist
and person, I deeply admire. And I sure wish to continue working on interesting projects
with Brian throughout my career.
I have also been very fortunate to work with Prof. Athol Kemball. It was terrific fun to
do radio astronomy stuff with him, which brought back memories of my Master’s research.
My training with him in technical aspects of the field was also, in no small part responsible
for my successful application with my future employers. He has been extremely kind to me
and I hope to continue our collaboration in future.
Also, I have been privileged to have worked with two giants of particle physics and
astrophysics, Prof. Keith Olive and Prof. Scott Dodelson. My first paper with Prof. Olive
was a huge learning experience, and absolute delight and occasional fright to keep up with
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the lightning speed of thought and profound depth of understanding. Having read Prof.
Dodelson’s book on Modern Cosmology and papers, collaborating with him has been a
highlight of my Ph.D. work. Once, again it was a lot of fun in addition to being a huge
challenge to try to keep up with the high standards of research.
The support and encouragement of senior graduate students is very important to the
growth of a new graduate student. I have been very fortunate to have wonderful seniors who
have become great friends aside from being fine examples to look upto.
My friend and senior at UIUC, Rishi Khatri has been a huge influence and support even
before the time I reached UIUC. From helping me settle down in a new country to guiding
me about some research decisions with his characteristic candour, he has and continues to
be an inspiration and an excellent friend.
One of my most favourite persons who also belongs to the Fields research family and
has been a very good friend and a part of my research support system is Amy Lien. She
is without a doubt, a true protege of Brian, always being the nicest person and also an
excellent scientist. My collaboration and friendship with Amy has enriched my life as a
graduate student, and I cherish our friendship and collaboration.
Another member of the Fields research family, the prolific Vasiliki Pavlidou has been a
tremendous source of encouragement and inspiration. I have begun collaborating with Vaso
towards the end of my Ph.D. and I look forward to many papers together. I also, look
forward to her eminently enjoyable company as I head to Europe.
My life as a graduate student has been so very enjoyable due to the presence of good
friends in this very tightly knit department. Kuo-Chuan Pan, I-Jen, Hotaka, Yiran and
David have been wonderful friends and classmates. They have always been there for me
research and otherwise. I will always have fond memories of all the fun times we have had
together as a group and feel lucky to have made such good friends.
This department and its various members have been very kind and helpful to me at every
step of my six years of graduate life. Jeri, Mary-Margaret, Sandy, Bryan and Judy have been
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extremely helpful and made it so much easier for me to do my research with everything else
taken care of. I certainly owe them a lot for comfortable life as a graduate student. And
Jeri, of course, is the kindest and sweetest person on this planet ! I will miss her a lot along
with everyone else in the department.
A huge part of my support system in Urbana-Champaign, are my friends from outside
the astronomy department. I cherish my friendship with Vineeth, Sarang, Pragnyadipta,
Sandeep, Vineet, Anika and the quintessential scientist Ritoban Basu Thakur. They have
all been of tremendous help in me settling down in the US and shared an unforgettable
common experience. My times spent taking classes and discussing science with Rito or
RBT, is something that has helped me strive towards doing exciting science. He has made
me better at being a scientist. Sandeep has been the most influential, my mentor at physical
training which has made me a lot sharper at my work. Indeed their friendship has been
invaluable and the gain of a lifetime that I take back with me.
Finally, my family including my parents, grandparents and my wife Soma, has been
my ultimate support system that made it possible for me to stay away from them and yet
function with some degree of success. My parents have been instrumental in me pursuing
science while being in India, still not a very popular choice. However, their atypical choices
have inspired me to do the same. Its their constant support and encouragement in chasing
my ambition of being a physicist that has given me freedom to do what I want, the way I
want. My father, Maloy who did his Ph.D. in Chemistry has always been a great source of
support, inspiration and encouragement taking a lot of detailed interest in my career. The
pride my parents take in my accomplishments is the biggest reward I gain.
My wife Soma has been the ideal partner in every possible way. It is testament to her
character, that she has never made us feel the geographical distance and has been central
to my growth as a scientist and person. She has helped me maintain the balance of the
personal and the professional in my life, impeccably. Being a researcher herself, Soma has
shown great patience and understanding throughout my Ph.D. and given a lot of importance
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to the accomplishment of my long and short term goals. At the same time, she has kept
me honest with my feet on the ground at all times. And therefore, when in doubt both
professionally and personally, she is the one I go to. I look forward to my future with her
in Europe. I hope, we can continue to support and improve each other as a scientist and a
person.
This work is a culmination of contributions from all these people, and particularly my
family. I dedicate my thesis to my family, particularly my beloved late grandfather who has
always had a lot of pride in my achievements.
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Table of Contents
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Primordial nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Cosmic gamma-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 High energy polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Radio Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 2 Resonant Destruction of Lithium During Big Bang Nucleosyn-thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Semi-analytical estimate of important reaction rates . . . . . . . . . . . . . 192.3 Systematic Search for Resonances . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 List of Candidate Resonances . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Narrow resonance solution space . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 A = 8 Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 A = 9 Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . . 332.4.3 A = 10 Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . 372.4.4 A = 11 Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Reduced List of Candidate Resonances . . . . . . . . . . . . . . . . . . . . . 492.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 3 Inverse-Compton Emission from Star-forming Galaxies . . . . . 553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Order-of-Magnitude Expectations . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Targets: Interstellar Photon Fields . . . . . . . . . . . . . . . . . . . . . . . 633.4 Projectiles: Cosmic-Ray Electron Source and Propagation . . . . . . . . . . 673.5 Inverse Compton Emission from Individual Star-Forming Galaxies . . . . . 72
3.5.1 Inverse Compton Emissivity of a Star-Forming Galaxy . . . . . . . . 723.5.2 Total Inverse-Compton Luminosity from a Single Galaxy . . . . . . . 75
3.6 Results: Inverse Compton Contribution to the Extragalactic Background . . 803.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 4 Statistics of gamma-ray sources . . . . . . . . . . . . . . . . . . . 934.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
viii
Chapter 5 High Energy Polarisation . . . . . . . . . . . . . . . . . . . . . . . 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Degree of polarisation of blazar . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 Minimum Detectable Polarisation (MDP) - Detection sensitivity of telescopes 1045.4 Flaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.5 Indirect constraints on hadronic emission . . . . . . . . . . . . . . . . . . . . 1135.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 6 Radio Supernovae in the Great Survey Era . . . . . . . . . . . . 1176.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Radio Properties of Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.1 Radio Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . . 1206.2.2 Radio Type Ia Supernovae . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Next-Generation Radio Telescopes: Expected Sensitivity . . . . . . . . . . . 1236.4 Radio Supernovae for SKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.1 Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.2 Type Ia Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 Radio Survey Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 1326.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter 7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 1377.1 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.1 The Narrow Resonance Approximation . . . . . . . . . . . . . . . . . . . . . 142
A.1.1 Narrow Normal Resonances . . . . . . . . . . . . . . . . . . . . . . . 143A.1.2 Narrow Subhreshold Resonances . . . . . . . . . . . . . . . . . . . . . 144
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146B.1 The energy loss rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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Chapter 1
Introduction
Nuclear and particle astrophysics comprises of applying standard principles of nuclear and
particle physics to astronomical systems and making predictions for observation. Also, ex-
treme environments in astronomical systems often serve as a probe for re-evaluating fun-
damental principles of physics. The research in this thesis comprises of both these aspects
of nuclear and particle astrophysics. One of the first steps in exploring new fundamental
physics with astrophysical probes, is to attain a systematic and comprehensive understanding
of the standard astrophysical processes and sources. Only after modeling these astrophysical
“foregrounds” can we confidently constrain models beyond the Standard Model of particle
physics and cosmology. With this broad vision in mind, I have worked on various projects
in the areas of primordial nucleosynthesis, the cosmic gamma-rays, high energy polarisation,
forecasting radio supernovae discovery, etc.
1.1 Primordial nucleosynthesis
As a part of the first topic, I have worked on the “Cosmological Lithium Problem”. The big
bang produces light elements namely H, He, Li and isotopes and isobars. And comparison of
theoretical predictions about the production of these elements with observations constrains
big bang cosmology (Wagoner et al., 1967a; Kolb & Turner, 1990; Cyburt, 2004a; Steigman,
2007, etc.). Big bang nucleosynthesis represents the competition between rate of nuclear,
weak and electromagnetic reactions, Γ and the expansion rate of the universe, or the Hubble
rate, H = a/a. Here a is the scale factor by which is unity at the present moment. The
1
reactions proceed to produce and destroy nuclei and achieve nuclear statistical equilibrium
between the various production and destruction reactions. This depends on the temperature
of the universe and also the baryon nb and photon densities nγ . In equilibrium, the rate of
interactions per particle exceed the expansion rate as
Γ = nσv ≫ H (1.1)
where σ is the interaction cross-section. Nucleosynthesis is quantified by through abun-
dances. And abundance of an element i is defined in terms of mole fraction Yi = ni/nb, with
respect to the baryon density of the universe at the time. The abundances are defined to be
the mole fractions, given relative to hydrogen, as
Yi
YH
=ni/nb
nH/nb
(1.2)
The dilution in the amount of these elements due to the expansion is precisely calculable
and a common factor for all nuclei. Yi is thus independent of this dilution factor. More
quantitatively the nuclear statistical equilibrium for a nucleus with abundance Y can be
quantified in terms of the abundances of projectiles, Yproj that produce it at a rate Γprod and
destruction rates, Γdest
dY
dt= nb(
∑
prod
YprojΓprod −∑
dest
Y Γdest)
=>a
Yeq
dY
da= − Γ
H
((
Y
Yeq
)2
− 1
)
(1.3)
and the equilibrium abundance corresponding to Γ ≫ H is
Yeq =
∑
YprojΓprod∑
Γdest
(1.4)
2
These abundances are functions of the temperature and also the baryon-to-photon ratio. Now
as the universe undergoes accelerated expansion, the expansion rate exceeds the production
and destruction reactions. As a result of this, different reactions fall out of equilibrium, and
the abundances set by those reactions are held fixed aside from the obvious dilution due to
expansion. This is falling out of equilibrium is known as “freeze-out” and the condition is
given as
Γ ≈ H (1.5)
Once, the expansion rate, H far exceeds reaction rate, Γ the abundances, Y are fixed. These
frozen out abundances depend only upon the baryon-to-photon density ratio represented by
η = nb/nγ ∝ Ωb/Ωγ. Here the Ω′s are ratio of matter-energy density of that component to
the critical density required to close the universe i.e. keep it from expanding forever.
Nucleosynthesis occurs at MeV energies and finishes within the first 3 minutes post the
big bang. It begins with protons and neutrons combining to produce deuterium. Weak inter-
actions interconvert neutrons and protons and when expansion exceeds the weak interaction
scale, the weak freezeout occurs setting the neutron to proton ratio(
np
)
∼ 1/6. The neutron
decays with a mean lifetime of ∼ 882 seconds. So a further decay of neutrons reduces this
ratio to 1/7, and this is when nucleosynthesis begins at the scale of the weak freezeout i.e.
at MeV temperatures. However, in addition to nucleons, there are about a billion times as
many photons in the radiation dominated universe at the time. This prevents neutrons and
protons from combining effectively. As a result, deuterium is photo-disintegrated as soon as
it is produced. This continues till about 100 seconds, at which point the universe cools to be-
low the binding energy of deuterium and “bottleneck” is released. Nucleosynthesis proceeds
to produce He and Li. The abundances of these elements after their respective freezeouts
are functions of the baryon-to-photon ratio alone. Therefore, observations of the elemental
abundances put constraints on this baryon-to-photon ratio other than parameters like the
3
gravitational constant, number of relativistic species, etc. that are fixed within the Standard
Model of nuclear and particle physics and cosmology. And through the baryon-to-photon
ratio or simply the baryon density, these abundances put constraints on big bang cosmology
(Kolb & Turner, 1990).
Further advances in the measuring the cosmic microwave background (CMB) have made
it a more precise baryometer than big bang nucleosynthesis (BBN) (Kneller et al., 2001;
Cyburt et al., 2003; Cuoco et al., 2004). Using the CMB value of Ωb, it possible to make
predictions for the elemental abundances of elements deuterium, tritium, helium and lithium
and isobars and isotopes. And these can be compared with observations. Thus, within the
framework of the Standard Model of particle physics and cosmology, there is no room /
freedom for adjusting the theoretical predictions of light elemental abundances.
Ideally, one would hope to observe these elements in pristine sites unpolluted by post
BBN- processing. However, in practice the observations are often affected by non-primordial
processes as stellar nucleosynthesis (For eg., Korn et al., 2006), cosmic rays and other as-
tronomical processes. Constraints from CMB are either weak as in the case of 4He or
unattainable due to the very nature of recombination. So the observations need to be per-
formed at the metal poor astronomical sites, minimally polluted by the heavier elements
or “metals” produced by aforementioned processes. 4He is observed in metal poor compact
dwarfs (Izotov et al., 1999). D is constrained from quasar absorption lines (O’Meara et al.,
2006; Fumagalli et al., 2011). 3He is observed in Galactic HII regions (Bania et al., 2002).
Observations of 7Li are performed at multiple sites as metal poor halo field stars (For e.g.
Spite & Spite, 1982; Ryan et al., 2000; Asplund et al., 2006) , globular cluster stars (Bonifa-
cio et al., 2002; Bonifacio, 2002; Gonzalez Hernandez et al., 2009) and more recently, in the
Small Magellanic Cloud (Howk et al., 2012). The 7Li observations in metal poor halo stars
show signs of a plateau called the “Spite Plateau” with a small scatter consistent with ob-
servational uncertainties. This suggests that the abundance is primordial. The observations
(Ryan et al., 2000) however, show an abundance 3-4 times less (Cyburt et al., 2008) than
4
predicted by standard BBN theory. This is known as the cosmological lithium problem.
One possible solution to the lithium problem is to look at nuclear reactions destroying
7Li or 7Be. 7Be is important as it is predominantly in this isobaric form that mass 7 nuclei
are formed in the early universe. Being β unstable it undergoes decay to produce the lithium
that is ultimately observed. The theoretical computation of the elemental abundances are
made by evolving the nuclear reactions both production and destruction, with the expansion
of the universe (Wagoner et al., 1967a; Smith et al., 1993). This in principle requires a
complete database of nuclear reactions whose properties are measured experimentally. In
practice, there are 12 important reactions including the neutron decay Smith et al. (1993).
Nuclear reactions that compete with these reactions that were either missed before or have
newer measurements will alter the abundances. Also, alternate nuclear channels, defined
by the quantum state of the reactants in these reactions, which were missed should also be
accounted for.
In Chakraborty et al. (2011), a systematic and complete study of these effects were made.
The 7Li and 7Be production reactions are very well measured leaving very little scope for
further exploration. So only destruction reactions need to be considered. It was found from
a semi-analytic estimate that only nuclear resonances were competitive enough with the 12
key reactions to alter the 7Li or 7Be abundances. This is based on the equilibrium abundance
in eq. 1.4). Thereby, a complete, systematic study of such resonances destroying 7Li and 7Be
down to observed abundances was made. The existence of such a resonance would represent
a nuclear solution to the cosmological 7Li problem that doesn’t require any new physics
beyond the Standard Model of particle physics and cosmology.
In doing so, data on resonances was used from resources such as NACRE 1, TUNL 2,
NNDC 3, etc. to construct resonant reaction rates to expand the nuclear reaction network
(Caughlan & Fowler, 1988; Angulo et al., 1999), in addition to the existing non-resonant
1http : //pntpm3.ulb.ac.be/Nacre/barre database.htm2http : //www.tunl.duke.edu/nucldata/3http : //www.nndc.bnl.gov/
5
contributions. Resonant energies and widths were parametrised where data was not available
and constrained the parameter space where 7Li acquires observed values. In doing so I
also, computed the amounts of heavier elements like 9Be are not overproduced by these
destruction reactions. Three candidate resonant levels for the initial states 7Be + d, 7Be + t
and 7Be+3He were found to have inelastic exit channels including p,3He and 4He which could
not be immediately dismissed based on the data on channel energies and widths available
then. However, owing to large Coulomb barriers, they all needed large channel radii (a > 10
fm) for solving the 7Li problem which makes this very difficult. More recently, 7Be + d
channel was ruled out explicitly by new data. This strongly suggests that systematically
accounting for standard nuclear physics cannot by itself solve the 7Li problem. As a result,
the 7Li problem may point to physics beyond the Standard Model of particle physics and
cosmology. These results along with work of others (Cyburt & Pospelov, 2012; Boyd et al.,
2010a) have prompted experiments (Charity et al., 2011; Kirsebom & Davids, 2011; O’Malley
et al., 2011) to look for these resonances. And these experiments have ruled out the best
candidate(s) as solutions to the lithium problem.
In absence, of this solution, it is a possibility that new physics beyond the Standard Model
has to be invoked in order to find a theoretical solution to the lithium problem. Several ways
of destroying exist from dark matter decay (Jedamzik, 2004; Jedamzik & Pospelov, 2009;
Ellis et al., 2005, etc.), variation of fundamental constants like quark masses (Dmitriev et al.,
2004), neutron lifetime, (Coc et al., 2007, etc.), etc., to formation of bound states (Cyburt
et al., 2006, etc.). Another class of solutions lies in stellar depletion of lithium in stars via
rotational mixing, turbulence, diffusion, etc (For e.g. Melendez et al., 2010b). Aside from
ways of destroying lithium, a reinterpretation of the observations due to an improved model
of stellar atmospheres may modify and reconcile the observed abundance with the theoretical
predictions (For e.g. Melendez et al., 2010a)
6
1.2 Cosmic gamma-rays
High energy radiation provides a natural, astrophysical probe to the some of the highest
energy phenomena and particles in the universe, all the way upto Planck scales. Thus,
cosmic gamma rays emitted by various astronomical sources both cosmological and local are
a powerful probe to the high energy universe. With the advances in gamma-ray astronomy
and the current state of the art with Fermi -LAT, it has become possible to investigate
individually interesting sources as well as populations of sources such as star forming galaxies
(Abdo et al., 2010f,a, etc.,), active galaxies or active galactic nuclei (Nolan et al., 2012a)
(AGN), compact objects like gamma-ray bursts (Fermi-LAT Collaboration, 2013), ordinary
black holes, etc. Now, in addition to known sources, gamma-rays are probes to the unknown
and the exotic. Just as BBN, cosmic gamma-rays too are an indirect probe of particle dark
matter (Ando et al., 2007; Regis & Ullio, 2008; Ackermann et al., 2012b). Dark matter decays
and annihilations produce gamma-rays at various energies and studying their properties
gives clues about dark matter models. However, it is crucial as in the case of BBN to
eliminate all the Standard Model foregrounds that could mimic or distort the dark matter
signature. Or alternately, we must model all the standard astrophysical sources that can
explain the data before invoking physics beyond Standard Model as would be the case for
dark matter. Therefore, even in the study of the dark matter, it is important to study
the known sources mentioned before. Various properties of gamma-ray sources such as
source spectra (Stecker & Venters, 2011; Lacki et al., 2012; Fields et al., 2010; Ackermann
et al., 2012c), flux distribution and statistics (Abdo et al., 2010l; Malyshev & Hogg, 2011),
anisotropies (Ackermann et al., 2012a; Hensley et al., 2010), etc. have been studied in order
to firstly, understand the distinctive signatures of the various astrophysical sources producing
gamma-rays. This is achieved by studying these properties of the resolved sources. Secondly,
EGRET (Sreekumar et al., 1998) and now Fermi -LAT (Abdo et al., 2010j) has helped
confirm the existence of a diffuse, extra-Galactic gamma-ray background (EGB). Efforts are
on to try to identify the origin of this diffuse EGB. Blazars or AGNs with their jets pointed
7
towards the observer, have been identified as the being the dominant source class amongst
the resolved sources. It is therefore natural to determine their contribution to the unresolved
sources or diffuse background. It is clear from the results of various groups including ours
(For eg Chakraborty & Fields, 2012; Stecker & Venters, 2011; Malyshev & Hogg, 2011)
that the diffuse background is multi-component. It is unlikely to be dominated by AGNs
due to guaranteed contribution from other sources mainly star forming galaxies Pavlidou &
Fields (2002); Fields et al. (2010). Despite knowing this, there is still debate on the exact
contribution of the star forming galaxies and AGNs, the two guaranteed sources of the EGB.
It is important to sharpen the constraints on their relative contributions.
As individuals objects, star-forming galaxies are laboratories of cosmic ray physics with
X-ray and gamma-ray telescopes being the apparatus. High energy radiation is produced in
sites of supernova remnants (SNR) due to cosmic ray interactions. Therefore, the gamma-
rays produced for instance are cosmic gamma rays. Multiwavelength observations provide
evidence of cosmic ray acceleration in sites of SNR upto energies of ∼ 1015 eV (Uchiyama
et al., 2007; Helder et al., 2009, etc.,). The cosmic ray flux φCR is the proportional to the
supernova rate. The supernova rate in turn, is proportional to the massive star-formation
rate, which ties it to the gamma-ray luminosity in sites of SNR in star-forming galaxies.
There exist direct connections between cosmic gamma rays and star-formation. This con-
nection is manifested in the context of the gamma-ray luminosities of individual galaxies
as well as the diffuse extra-Galactic gamma ray background (EGB).Star-forming galaxies
produce gamma-rays by cosmic ray interactions with the interstellar medium (ISM). And
therefore, the gamma-ray emissivity, qγ is given in terms of the cosmic ray flux φCR and the
ISM density nISM as,
qγ ∼ φCRσintnISM (1.6)
Gamma ray luminosities of individual star-forming galaxies are correlated to the galactic
8
star-formation rate and the cosmic star-forming background to the cosmic star-formation
rate. The exact nature of the correlation depends on the emission mechanism, hadronic or
leptonic and their efficiencies.
Star-forming galaxies from the Local Group have been detected at GeV energies by Fermi
-LAT (Abdo et al., 2010g). In addition to the Milky Way this includes normal galaxies
M 31(Abdo et al., 2010g), Large Magellanic Cloud (LMC) (Abdo et al., 2010h), Small
Magellanic Cloud (SMC) (Abdo et al., 2010c) and starbursts M 82 and NGC 253 (Abdo
et al., 2010b). M33 is very likely to be detected by Fermi -LAT in the upcoming years
(Abdo et al., 2010g). It is believed that the dominant emission mechanism for Milky Way
type galaxies GeV gamma-ray production is hadronic or due to interactions of cosmic ray
protons with interstellar gas. (neutral, ionised and molecular hydrogen) to produce neutral
pion that decay in flight into gamma rays. The gamma-ray emission can be viewed in terms
of collisions between projectiles i.e. the cosmic ray protons with the targets i.e. the ISM
protons.
pCR + pISM → p+ p+ π0 π0 → γγ (1.7)
However, in some cases as in the Magellanic Clouds, this is not clear as the gas and the
gamma rays show very little spatial correlation. The next most important process of pro-
ducing gamma-rays is inverse-Compton scattering of cosmic ray electrons off the interstellar
radiation field. Here the projectiles are the cosmic ray electrons and the targets are the ISRF
photons.
e−CR + γISRF → e− + γHE (1.8)
In order to determine the exact contribution of star forming galaxies, it is important
to study the individual galaxies in detail as only the handful mentioned above have been
detected in gamma-rays Abdo et al. (2010g) in contrast the vast majority of AGNs. There-
9
fore, while for AGNs a statistical understanding of the gamma-ray emission suffices, for star
forming galaxies knowing details of the emission mechanisms in these handful are useful
to extrapolate to get their cosmological emission. Emission mechanisms of the individual
galaxies as well as the number density of these galaxies differ depending on whether they are
a starburst that has heightened periods of star formation activity or not. The ones which
dont have this starburst activity are called normal galaxies. Pavlidou & Fields (2002); Fields
et al. (2010), computed a model for the hadronic emission due to neutral pion decay from
normal star forming galaxies. This involves computation of the projectile or cosmic ray flux
in terms of the star formation rate based on the aforementioned correlation and normalised
to the Milky Way. The target density is the interstellar hydrogen density in various forms
i.e. neutral, molecular and ionised. The relevant quantity here is the surface gas density that
in turn is related to the surface density of star formation Kennicutt (1998). This implies
that the gamma-ray luminosity Lγ is a non-linear function of the star formation rate ψ. This
is a critical result from Pavlidou & Fields (2002); Fields et al. (2010) that sets the overall
scaling relation between the gamma-ray luminosity and the star formation rate. The shape
of course, is very well defined. A pion decaying at rest would have a delta function peak
at the rest mass of 67.5 MeV. However, the pion decays in flight. Furthermore, Galactic
rotation produces dispersion, and the peak becomes a bump. For external galaxies at cos-
mic distances, there is a redshifting as well. Using Hα as a tracer for these normal star
forming galaxies, the distribution function is constructed. The intensity is then computed
as the line of sight integral of product of the distribution as well as the luminosity of a single
galaxy. Fields et al. (2010) find that the cosmological contribution of the normal galaxies
due to neutral pion decay is significant and could in fact dominate the extra-Galactic diffuse
background upto around 10 GeV. The LAT range of course, extends to around 300 GeV.
Given that the inverse-Compton contribution from normal galaxies is also guaranteed
and the next most important, this contribution was computed in Chakraborty & Fields
(2012). Also, since, the inverse Compton spectral shape is flatter at higher energies it
10
could apriori be significant at energies above 10 GeV. This provides strong motivation to
perform this computation. This leptonic contribution has cosmic ray electrons as projectiles
instead of protons. Once again as for protons, the cosmic ray electron flux scales as the
star formation rate, being accelerated in the same SNR as protons. Their efficiency could
be different though. Electrons propagate differently from the protons. Radiative losses are
significant for electrons, unlike protons that are heavier. Also, while electrons diffuse they
do not escape the galaxy whereas proton losses in the normal galaxies are dominated by
escape. The radiative energy lost by electrons is equal to the energy output in photons at
various energies depending on loss mechanism. This is known as calorimetry. For instance,
synchrotron losses produce photons from radio all the way to X-rays. Inverse Compton losses
produce some X-rays and certainly gamma-rays. And thus the gamma-ray IC emission is
basically the fraction of the total energy lost in IC. As a result the gamma-ray luminosity
depends not the the background seed photon or ISRF density, but fraction of IC. Once
again from the luminosity and the Hα distribution the intensity of cosmological emission is
computed.
1.3 High energy polarisation
In section 1.2, gamma-ray properties of star forming galaxies and AGNs discussed include
the source intensity, spectra, flux distribution, etc. A property with tremendous scientific
potential in gamma-ray and even X-ray astronomy that is being explored more recently
(For e.g. McNamara et al., 2009; Krawczynski, 2012; Weisskopf et al., 2010) is polarisation.
Measurement of polarisation at X-ray and even soft gamma-rays is very challenging. But
several experiments in various stages of operation, design and commission are about to change
this and open a new window of opportunities in investigating extreme objects like AGNs,
GRBs, black holes in general, neutron stars, etc (McNamara et al., 2009; Krawczynski,
2012; Gotz et al., 2009). Some of the above sources are either established as or candidates
11
for particle accelerators all the way upto nearly the ankle of the cosmic ray spectrum (∼
1019eV). HIgh energy polarisation will probe of emission mechanisms that in turn will probe
acceleration mechanisms and thereby test the particle accelerators. Magnetic fields can
lead to polarised emission. And therefore, polarisation measurements will help understand
magnetic field structure surrounding the astronomical sources.
Lorentz invariance is one of the treasured symmetries in the Standard Model of particle
physics and cosmology. But certain particle physics and gravity models have Lorentz invari-
ance violation built into them. Gamma rays can be used to probe these models in many ways
(Coleman & Glashow, 1999; Ellis et al., 2000) One of the ways is to measure vacuum bire-
fringence, or modification of the dispersion relation of light due to Lorentz violating terms
in the underlying Lagrangian. Thus, putting constraints on these terms puts constraints
on Lorentz violation. This modified dispersion relation leads to a rotation of polarisation
during propagation of the photons. INTEGRAL / IBIS observations of the polarisation from
the prompt emission from the GRB041219A has put a strong limit on the cubic correction
term to the standard dispersion relation (Laurent et al., 2011)limiting the LIV parameter to
< 1.110×10−14, an improvement of 4 orders of magnitude over the previous limit. This limit
is inversely proportional to the distance, which suggests that polarisation measurements of
higher redshift GRBs can in principle improve the limits further.
In particular, polarised X-rays and gamma-rays from AGNs have the potential to reveal a
lot of different aspects on AGN physics. AGNs are some of the most luminous yet mysterious
astronomical objects in the universe. Their particle and radiative emissions are powered by a
central supermassive black hole accreting matter. Part of the gravitational energy associated
with accretion is converted into energy of particles such as cosmic rays and neutrinos and
high energy radiation like X-rays and gamma-rays. Thus, these particles and radiation are
messengers of the extreme astrophysical conditions in the core of active galaxies. Blazars are
distant AGNs where the observer’s line of sight is along the jet axis, i.e. the observer looks
down into the jets. Various properties of the radiation from blazars like the overall inten-
12
sity, spectrum, variability have been studied with multiwavelength observations. However,
polarisation particularly at high energies has received much less attention. However, with
numerous X-ray and soft gamma-ray polarimeters at various stages of planning, design and
operation and studies of optical / FIR polarisation properties of blazars underway, this is a
perfect time to calculate high energy polarisation from blazars. High energy polarisation is
a key ingredient in the multimessenger, multiwavelength understanding of blazars.
1.4 Radio Supernovae
Yet another piece in this gamma-ray - star formation connection are core collapse SNe.
Core collapse SNe are result of death of massive stars. And hence, the core collapse SNe
rate traces the massive star formation rate. In fact, they are in direct proportion to each
other. Thus building up a statistically significant sample of core-collapse SNe constitutes
an independent probe of the star formation rate. Core collapse SNe have been observed
to emit radio. Over 50 of them have been discovered with the current generation of radio
telescopes. However, with future generations of radio surveys that represent more than an
order of magnitude improvement in sensitivity, it is possible to discover many more core
collapse SNe. This discovery potential is explored in Lien et al. (2011). Radio has the
capability to go deeper than optical in discovering supernovae. The large optical telescopes
such as the Large Synoptic Survey Telescope (LSST) will detect a plethora of core-collapse
supernovae out to redshift 1. However, it is estimated by Mannucci et al. (2007) that
∼ 60% of core collapse supernovae maybe missed in optical surveys due to increased dust
obscuration at high redshift. In our work in Lien et al. (2011) , it was shown that future
generations of radio telescopes in survey mode such as SKA and its precursors will detect
a large fraction of core-collapse SNe as predicted from the cosmic massive star-formation
rate in Horiuchi et al. (2009a) out to redshift 5. Also, future radio telescope arrays could
be trained with simultaneous or triggered with preceding optical detections at low redshift.
13
And using this knowledge about radio supernovae at low redshift, they could be used for
independent discoveries at high redshift. Also, there is a potential shift in the detection
strategies from mainly targeted PI driven searches to untargeted, synoptic surveys capable
of detecting radio supernovae automatically.
The radio emission comes from the interaction of the wind with the circumstellar medium.
So other simply building a sample, naturally, the radio emission is a probe of the models of
the circumstellar medium and its various parts Chevalier (1982a,b).
About 25% of core-collapse SNe are Type 1bc Li et al. (2011a), which are related to
long GRBs. According to our estimate with very conservative assumptions about fraction
of radio emitting Ibc SNe, we expect SKA to make ∼ 130 unbiased, untargeted detections,
with ∼ 20 connected with GRBs. Aside from these, exotic transients whose properties do
not match those of known transient sources are yet another discovery pool.
A synoptic survey in radio wavelengths will be crucial in many fields of astrophysics,
particularly for RSNe, but also for other transients. It will bring the first complete and
unbiased RSN sample and systematically explore exotic radio transients. SKA will be ca-
pable of performing such an untargeted, possibly automated survey with its unprecedented
sensitivity. Our knowledge of supernovae will thus be firmly extended into the radio and to
high redshifts.
14
Chapter 2
Resonant Destruction of LithiumDuring Big Bang Nucleosynthesis
Abstract
1 We explore a nuclear physics resolution to the discrepancy between the predicted standard
big-bang nucleosynthesis (BBN) abundance of 7Li and its observational determination in
metal-poor stars. The theoretical 7Li abundance is 3 - 4 times greater than the observational
values, assuming the baryon-to-photon ratio, ηwmap, determined by WMAP. The 7Li problem
could be resolved within the standard BBN picture if additional destruction of A = 7 isotopes
occurs due to new nuclear reaction channels or upward corrections to existing channels. This
could be achieved via missed resonant nuclear reactions, which is the possibility we consider
here. We find some potential candidate resonances which can solve the lithium problem and
specify their required resonant energies and widths. For example, a 1− or 2− excited state of
10C sitting at approximately 15.0 MeV above its ground state with an effective width of order
10 keV could resolve the 7Li problem; the existence of this excited state needs experimental
verification. Other examples using known states include 7Be + t → 10B(18.80 MeV), and
7Be+d→ 9B(16.71 MeV). For all of these states, a large channel radius (a > 10 fm) is needed
to give sufficiently large widths. Experimental determination of these reaction strengths is
needed to rule out or confirm these nuclear physics solutions to the lithium problem.
1This chapter is previously published in The Physical Review D as Chakraborty, N., Fields, B. D., &Olive, K. A. 2011, Phys. Rev. D, vol. 83 pp. 63006. This chapter matches the published version aside fromsuperficial modifications to references.
15
2.1 Introduction
Primordial nucleosynthesis continues to stand as our earliest probe of the universe based
on Standard Model physics. Accurate estimates of the primordial abundances of the light
elements D, 4He and 7Li within standard Big Bang Nucleosynthesis (BBN) Cyburt et al.
(2001); Coc et al. (2004a); Cyburt et al. (2003); Cyburt (2004b); Cyburt et al. (2008) are
crucial for making comparisons with observational determinations and ultimately testing the
theory. Primordial abundances are also a probe of the early universe physics Cyburt et al.
(2005). Currently, the theoretical estimates of D and 4He match the observational values
within theoretical and observational uncertainties Cyburt et al. (2003, 2008) at the baryon-
to-photon ratio determined by the 7-year WMAP data, ηwmap = 6.19±0.15×10−10 Komatsu
et al. (2011). In contrast, the theoretical primordial abundance of 7Li does not match the
observations.
At ηwmap, the predicted BBN abundance of 7Li is2 Cyburt et al. (2008)
(
7Li
H
)
BBN
=(
5.12+0.71−0.62
)
× 10−10. (2.1)
The observed 7Li abundance is derived from observations of low-metallicity halo dwarf stars
which show a plateauSpite & Spite (1982) in (elemental) lithium versus metallicity, with a
small scatter consistent with observational uncertainties. An analysis Ryan et al. (2000) of
field halo stars gives a plateau abundance of
(
Li
H
)
halo⋆
= (1.23+0.34−0.16) × 10−10. (2.2)
However, the lithium abundance in several globular clusters tends to be somewhat higher Boni-
facio (2002); Gonzalez Hernandez et al. (2009), and a recent result found in Gonzalez
Hernandez et al. (2009) gave 7Li/H = (2.34 ± 0.05) × 10−10. Thus the theoretically es-
2Note that the 7Li abundance reported here differ slightly from that given in Cyburt et al. (2008),primarily due to the small shift in η as reported in Komatsu et al. (2011).
16
timated abundance of the isobar with mass 7 (7Be+7Li) is more than the observationally
determined value by a factor of 2.2 - 4.2 Cyburt et al. (2008), at ηwmap. Relative to the
theoretical and observational uncertainties, this represents a deviation of 4.5-5.5 σ.
This significant discrepancy constitutes the “lithium problem” which could point to lim-
itations in either the observations, our theoretical understanding of nucleosynthesis, or the
post-BBN processing of lithium.
On the theoretical front, strategies which have emerged to approach the lithium problem
broadly either address astrophysics or microphysics. On the astrophysical side, one might
attempt to improve our understanding of lithium depletion mechanisms operative in stellar
models Korn et al. (2006). This remains an important goal but is not our focus here.
The microphysical solutions to the lithium problem all in some way change the nuclear
reactions for lithium production in order to reduce the primordial (or pre-Galactic) lithium
abundance to observed levels. Some of these work within the Standard Model, focussing on
nuclear physics, in particular the nuclear reactions involved in lithium production. One ap-
proach is to attempt to utilize the experimental uncertainties in the rates Coc et al. (2004a);
Angulo et al. (2005); Cyburt et al. (2004); Boyd et al. (2010b). A second, related approach
is the inclusion of new effects in the nuclear reaction database such as poorly understood
resonance effects Cyburt & Pospelov (2012). Finally, it may happen that effects beyond
the Standard Model are responsible for the observed lithium abundance. For example, the
primordial lithium abundance can be reduced by cosmological variation of the fine structure
constant associated with a variation in the deuterium binding energy Coc et al. (2007), or
by the post-BBN destruction of lithium through the late decays of a massive particle in the
early universe Cyburt et al. (2010).
In this paper, we remain within the Standard Model, examining the possible role of
resonant reactions which may have been up to now neglected. The requisite reduction in the
7Li abundance can be achieved by either an enhancement in the rate of destruction of 7Li
or its mirror nucleus 7Be. This approach is more promising than the alternative of reducing
17
the production of 7Be and 7Li where the reactions are better understood experimentally
and theoretically Cyburt (2004a); Cyburt & Davids (2008); Ando et al. (2006), whereas the
experimental and especially the theoretical situation for A = 8 − 11 has made large strides
but still allows for surprises at the levels of interest to us Pieper et al. (2002).
The use of resonant channels is an approach that has paid off in the past in the context of
stellar nucleosynthesis. Fred Hoyle famously predicted a resonant energy level at 7.68 MeV
in the 12C compound nucleus which enhances the 8Be + α→ 12C reaction cross-section and
allows the triple alpha reaction to proceed at relatively low densities Hoyle (1954). Recently,
it was shown that there are promising resonant destruction mechanisms which can achieve
the desired reduction of the total A = 7 isotopic abundance Cyburt & Pospelov (2012). This
paper points to a resonant energy level at (E, Jπ) = (16.71 MeV, 5/2+) in the 9B compound
nucleus which can increase the rate of the 7Be(d,p)αα and/or 7Be(d, γ)9B and thereby reduce
the 7Be abundance. Here, we take a more general approach and systematically search for all
possible compound nuclei Tilley et al. (2004); Ajzenberg-Selove (1990) and potential resonant
channels which may result in the destruction of 7Be and/or 7Li.
Because of the large discrepancy between the observed and BBN abundance of 7Li, any
nuclear solution to the lithium problem will require a significant modification to the ex-
isting rates. As we discuss in the semi-analytical estimate in section 2.2, any new rate or
modification to an existing one, must be 2 - 3 times greater than the current dominant
destruction channels namely, 7Li(p, α)α for 7Li and 7Be(n, p)7Li for 7Be. As discussed in
Boyd et al. (2010b) and as we show semi-analytically in § 2.2, this is difficult to achieve
with non-resonant reactions. Hence, we will concentrate on possible resonant reactions as
potential solutions to the lithium problem. As we will show, there are interesting candidate
resonant channels which may resolve the 7Li problem. For example, there is a possibility of
destroying 7Be through a 1− or 2− 10C excited state at approximately 15.0 MeV. The energy
range between 6.5 and 16.5 MeV is currently very poorly mapped out and a state near the
entrance energy for 7Be + 3He could provide a solution if the effective width is of order 10
18
keV. We will also see that these reactions all require fortuitously favorable nuclear param-
eters, in the form of large channel radii, as also found by Cyburt and Pospelov Cyburt &
Pospelov (2012) in the case of 7Be+d. Even so, in the face of the more radical alternative of
new fundamental particle physics, these more conventional solutions to the lithium problem
beckon for experimental testing.
The paper is organized as follows: First, we lay down the required range of properties of
any resonance to solve the lithium problem by means of a semi-analytic estimate inspired
by Mukhanov (2004); Esmailzadeh et al. (1991) in § 2.2. Then, in § 2.3, we list experimen-
tally identified resonances from the databases: TUNL 3 and NNDC 4, involving either the
destruction of 7Be or 7Li. Finally, the solution space of resonant properties, wherein the
lithium problem is partially or completely solved, is mapped for the most promising initial
states involving either 7Li or 7Be, by including these rates in a numerical estimation of the
7Li abundance. This exercise will delineate the effectiveness of experimentally studied or
identified resonances as well as requirements of possible missed resonant energy levels in
compound nuclei formed by these initial states. This is described in § 2.4. We note that
in our analysis, the narrow resonance approximation is assumed which may not hold true
in certain regions of this solution space. Our key results are pared down to a few resonant
reactions described in § 2.5. A summary and conclusions are given in § 2.6.
2.2 Semi-analytical estimate of important reaction
rates
Before we embark on a systemic survey of possible resonant enhancements of the destruction
of A = 7 isotopes, it will be useful to estimate the degree to which the destruction rates must
change in order to have an impact on the final 7Li abundance. The net rate of production
3http : //www.tunl.duke.edu/nucldata/4http : //www.nndc.bnl.gov/
19
of a nuclide i is given by the difference between the production from nuclides k and l and
the destruction rates via nuclide j, i.e. for the reaction i+ j → k + l. This is expressed
quantitatively by the rate equation Wagoner et al. (1967b) for abundance change
dni
dt= −3Hni +
∑
jkl
nknl〈σv〉kl − ninj〈σv〉ij, (2.3)
where ni is the number density of nuclide i,H is the Hubble parameter,∑
ij ninj〈σv〉ij are the
sum of contributions from all the forward reactions destroying nuclide i and∑
kl nknl〈σv〉kl
are the reverse reactions producing it. 〈σv〉 is the thermally averaged cross-section of the
reaction. The dilution of the density of these nuclides due to the expansion of the universe
can be removed by re-expressing eq. (2.3) in terms of number densities relative to the baryon
density Yi ≡ ni/nb, as,
dYi
dt= nb
∑
jkl
YkYl〈σv〉kl − YiYj〈σv〉ij . (2.4)
Using this general form, the net rate of 7Be production can be approximated in terms of the
thermally averaged cross-sections of its most important production and destruction channels
as
dY7Be
dt= nb (〈σv〉3HeαY3HeYα − 〈σv〉7BenY7BeYn) . (2.5)
Here, the reverse reaction rates of these production and destruction channels are neglected,
as they are much smaller than the forward rates at the lithium synthesis temperature. A
similar equation can be written down for 7Li. When quasi-static equilibrium is reached, the
destruction and production rates are equal. In this case, approximate values for new rates,
which can effectively destroy either isobar, can be obtained analytically.
At temperatures T ≈ 0.04 MeV, both 7Li and 7Be are in equilibrium Mukhanov (2004)
20
which gives,
〈σv〉3HeαY3HeYα = 〈σv〉7BenY7BeYn . (2.6)
Consider a new, inelastic 7Be destruction channel 7Be+X → Y +Z, involving projectile X.
This reaction will add to the right hand side of eq. (2.6) and shift the equilibrium abundance
of 7Be to a new value as follows,
Y new7Be ≈ 〈σv〉3HeαYαY3He
〈σv〉7BenYn + 〈σv〉7BeXYX≈ 1
1 +〈σv〉7BeXYX
〈σv〉7BenYn
Y old7Be . (2.7)
If the new reaction is to be important in solving the lithium problem, it must reduce the
7Be abundance by a factor of Y new7Be
/Y old7Be
∼ 3 − 4 . This in turn demands via eq. (2.7) that
〈σv〉7BeXYX/〈σv〉7BenYn ∼ 2 − 3, i.e., the rate for the new reaction exceeds that of the usual
n− p interconversion reaction rate. A similar estimate can be made for 7Li.
This reasoning would exclude non-resonant rates as they would be required to have
unphysically large astrophysical S-factors in the range of order 105−109 keV - barn depending
on the channel. Thus we would expect that only resonant reactions can produce the requisite
high rates. Possible resonant reactions are listed in the next section, whose key properties of
resonance strength, Γeff and energy, Eres, lie in appropriate ranges capable of achieving the
required destruction of mass 7.
Finally, we turn to 7Li destruction reactions, 7Li+X → Y +Z. Recall that at the WMAP
value of η, mass 7 is made predominantly as 7Be, with direct 7Li production about an order
of magnitude smaller. This suggests that enhancing direct 7Li destruction will only modestly
affect the final mass-7 abundance; we will see that this expectation is largely correct.5
With these pointers, the list in the next section is reduced and numerical analysis of the
5A subtle point is that normally, the mass-7 abundance is most sensitive to rate 7Be(n, p)7Li Smith et al.(1993). Of course, this reaction leaves the mass-7 abundance unchanged, but the lower Coulomb barrier for7Li leaves it vulnerable to the 7Li(p, α)4He reaction, which is extremely effective in removing 7Li. Thus, fora new, resonant 7Li destruction reaction to be important, it must successfully compete with the very large7Li(p, α)4He rate, and even then the mass-7 destruction “bottleneck” remains the 7Be(n, p)7Li rate thatlimits 7Li appearance. Thus we would not expect direct 7Li destruction to be effective. We will examine 7Lidestruction below, and confirm these expectations.
21
remaining promising rates is done.
2.3 Systematic Search for Resonances
In this section we describe a systematic search for nuclear resonances which could affect
primordial lithium production. We first begin with general considerations, then catalog the
candidate resonances. We briefly review the basic physics of resonant reactions to establish
notation and highlight the key physical ingredients.
2.3.1 General Considerations
Energetically, the net process 7Be + A → B +D must have Q+ Einit ≥ 0, where the initial
kinetic energy Einit ≃ T <∼ 40 keV is small at the epoch of A = 7 formation. Thus we in
practice require exothermic reactions, Q > 0. Moreover, inelastic reactions with large Q will
yield final state particles with large kinetic energies. Such final states thus have larger phase
space than those with small Q and in that sense should be favored.
Consider now a process 7Be + X → C∗ → Y + Z which destroys 7Be via a resonant
compound state; a similar expression can be written for 7Li destruction. In the entrance
channel 7Be + X → C∗ the energy released in producing the compound state is QC =
∆(7Be) + ∆(X) − ∆(Cg.s.), where ∆(A) = m− Amu is the mass defect. If an excited state
C∗ in the compound nucleus lies at energy Eex, then the difference
Eres ≡ Eex −QC (2.8)
determines the effectiveness of the resonance. We can expect resonant production of C∗
if Eres <∼ T . In an ordinary (“superthreshold”) resonance we then have Eres > 0, while a
subthreshold resonance has Eres < 0.
Once formed, the excited C∗ level can decay via some set of channels. The cross section
22
for 7Be +X → C∗ → Y + Z is given by the Breit-Wigner expression
σ(E) =πω
2µE
ΓinitΓfin
(E −Eres)2 + (Γtot/2)2(2.9)
where E is the center-of-mass kinetic energy in the initial state, µ is the reduced mass and
ω =2JC∗ + 1
(2JX + 1)(2J7 + 1)(2.10)
is a statistical factor accounting for angular momentum. The width of the initial state
(entrance channel) is Γinit, and the width of the final state (exit channel) is Γfin.
One decay channel which must always be available is the entrance channel itself. Obvi-
ously such an elastic reaction is useless from our point of view. Rather, we are interested
in inelastic reactions in which the initial 7Be (or 7Li) is transformed to something else. In
some cases, an inelastic strong decay is possible where the final state particles Y + Z are
both nuclei. Note that it is possible to produce a final-state nucleus in an excited state, e.g.,
C∗ → Y ∗+Z, in which case the energy release Q′C is offset by the Y ∗ excitation energy. This
possibility increases the chances of finding energetically allowable final states. Indeed, such
a possibility has been suggested in connection with the 7Be + d → 9B∗ → 8Be
∗+ p process
Cyburt & Pospelov (2012).
Regardless of the availability of a strong inelastic channel, an electromagnetic transition
C∗ → C(∗)+γ to a lower level is always possible. However, these often have small widths and
thus a small branching ratio Γfin/Γtot. Thus for electromagnetic decays to be important, a
strong inelastic decay must not be available, and the rest of the reaction cross section needs
to be large to compensate the small branching; as seen in eq. (2.9), this implies that Γinit be
large.
Note that in all charged-particle reactions, the Coulomb barrier is crucially important
and is implicitly encoded via the usual exponential Gamow factor in the reaction widths of
both initial and final charged-particle states. However, if the reaction has a high Q, the final
23
state kinetic energy will be large and thus there will not be significant final-state Coulomb
supression; this again favors final states with large Q. In addition, if the entrance or exit
channel has orbital angular momentum L > 0, there is additional exponential suppression,
so that L > 0 states are disfavored for our purposes.
With these requirements in mind, we will systematically search for resonant reactions
which could ameliorate or solve the lithium problem. We begin by identifying possible
processes which are
1. new resonances not yet included in the BBN code;
2. 2-body to 2-body processes, since 3-body rates are generally very small in BBN due
to phase space suppression as well as the relatively low particle densities and short
timescales;
3. experimentally allowed – in practice this means we seek unidentified states in poorly
studied regimes;
4. narrow resonances having Γtot <∼ T , which is around Γtot < 40 keV but we will also
consider somewhat larger widths to be conservatively generous.
5. relatively low-lying resonances with Eres <∼ few × T ∼ 100 − 300 keV, which are
thermally accessible; here again we err on the side of a generous range.
Once we have identified all possible candidate resonances, we will then assess their viability
as solutions to the lithium problem based on available nuclear data.
2.3.2 List of Candidate Resonances
As described above, we will explore the resonant destruction channels of both 7Li and 7Be.
Some of the potential resonances which might be able to reduce the mass 7 abundance to
the observed value were recently considered in Cyburt & Pospelov (2012). This analysis
eliminates several candidate resonances, leaving as genuine solutions only the resonance
24
related to the 7Be(d, γ)9B and 7Be(d, p)αα reactions and associated with the 16.71 MeV
level in the 9B compound nucleus. Here, we make an exhaustive list of possible promising
resonances that may be important to either 7Be or 7Li destruction channels. In order to
do so systematically and account for all possible resonances that may be of importance, we
study the energy levels in all possible compound nuclei that may be formed in destroying
7Be or 7Li, making extensive use of databases at TUNL and the NNDC 6.
The available 2-body destruction channels 7A+X may be classified byX = n, p, d, t, 3He, α,
and γ. Consequently, the compound nuclei that can be formed starting from mass 7 have
mass numbers ranging from A = 8 to A = 11, and the ones of particular interest are
8Li, 8Be, 8B, 9Be, 9B, 10Be, 10B, 10C, 11B and 11C. All relevant, resonant energy levels in these
compound nuclei that may provide paths for reduction of mass-7 abundance are listed in
Tables 2.1 – 2.10.
There are quantum mechanical and kinematic restrictions to our selection of candidates.
The candidate resonant reactions must obey selection rules. The partial widths for a channel,
which may be viewed as probability currents of emission of the particle in that channel
through the nuclear surface, are given as
ΓL(E) = 2 ka PL(E, a)γ2(a) (2.11)
where a is the channel radius and E is the projectile energy. Here k is the wavenumber of the
colliding particles in the centre-of-mass frame and γ2 is the reduced width, which depends on
the overlap between the wavefunctions inside and outside the nuclear surface, beyond which
the nuclear forces are unimportant. The reduced width, γ2 is independent of energy and has
a statistical upper limit called the Wigner limit given by Teichmann & Wigner (1952)
γ2 ≤ 3~2
2µa2, (2.12)
6http : //www.tunl.duke.edu/nucldata/,http : //www.nndc.bnl.gov/
25
The pre-factor of 32
is under the assumption that the nucleus is uniform and can change
to within a factor of order unity if this assumption changes. The Wigner limit depends
sensitively on the channel radius and thus varies with the nuclei involved. For the nuclei of
our interest, typical values of γ2 range from a few hundred keVs to a few MeVs.
In eq. (2.11), PL(E, a) is the Coulomb penetration probability for angular momentum L
and is a strong and somewhat complicated function of E and a. Thus, while the Wigner
limit sets a theoretical limit on the reduced width, the upper limit on the full width, ΓL(E),
depends on the values of PL(E, a) and is sensitive to the details of the resonant channel
being considered. In light of this complexity, our strategy is as follows. We evaluate the
ΓL(E) needed to make a substantial impact on the lithium problem. Then for the cases of
highest interest, we will compare our results with the theoretical limit set by the Coulomb
suppressed Wigner limit for those specific cases.
We also limit our consideration to two body initial states, with resonance energies Eres ≤
650 keV. The high resonance energy limit ensures that all possible resonances which may
influence the final 7Li abundance are taken into account, though many of the channels with
such high resonance energies will inevitably be eliminated. Excited final states have also been
considered in making this list. Different excited states of final state products are marked as
separate entries in the table, since each one has its own spin and therefore a different angular
momentum barrier. And thereby the significance of each excited state in destroying mass
7 is varied. Also, we usually eliminate the reactions with a negative Q-value except for the
7Li(d, p)8Li, 7Be(d, 3He)6Li, 7Be(d, p)8Be∗
(16.922 MeV) and 7Li(3He, p)9Be∗
(11.283 MeV)
as they are only marginally endothermic.
For a number of the reactions listed in these tables, 1-10, the total spin of the initial
state reactants is equal to that of the compound nucleus, which is equal to the total spin of
the products, with L = 0. However, for many reactions, angular momentum is required in
the initial and/or final state, which decreases the penetration probability and thereby the
width for that particular channel. In fact for some of these reactions, parity conservation
26
demands higher angular momentum which worsens this effect. However, we do not reject any
channel based on the angular momentum suppression of its width in these tables. Later we
will shortlist those resonant channels which are not very suppressed and indeed potentially
effective in destroying mass 7.
The reactions of interest are listed in increasing order of the mass of the compound nuclei
formed. The particular resonant energy levels of interest Eex and their spins are listed in the
table. In general, different initial states involving 7Li and 7Be can form these energy levels
and so all these relevant initial states are listed. For each one, the various final product
states for an inelastic reaction are enumerated. Again, each of the final state products can
also be formed in an excited state. These excited states must have lower energy than the
initial state energies for the reaction to be exothermic. In addition spin and parity must
be conserved. Enforcing these, the minimum final state angular momenta Lfin are evaluated
from the spin of the resonant energy level and are listed in the tables. The total widths of
the energy levels are listed whenever available. The partial widths of the different channels
including the elastic one, out of each energy level are also listed.
We adopt the narrow resonance approximation to evaluate the effect of these resonances
and either retain or dismiss them as potential solutions to the lithium problem. Some of
the partial widths or limits on them are high enough that they easily qualify to be broad
resonances. This implies that the narrow resonance formula used to see their effect is not
precise, but still gives a rough idea of whether the resonance is ineffective or not.
Our expression for thermonuclear rates in the narrow resonance approximation is ex-
plained in detail in Appendix A.1, and is given by
〈σv〉 = ω Γeff
(
2π
µT
)3/2
e−|Eres|/T f (2Eres/Γtot) (2.13)
This rate is controlled by two parameters specific to the compound nuclear state: Eres and
Γeff . Here Eres is given in eq. (2.8), and measures the offset from the entrance channel and
27
the compound state. The resonance strength is quantified via
Γeff =ΓinitΓfin
Γtot(2.14)
with Γinit and Γfin being the entrance and exit widths of a particular reaction, and Γtot the
sum of the widths of all possible channels. Of these widths, the smaller of Γinit and Γfin along
with Γtot are listed in the table above. The resonance strength, Γeff ≈ Γinit, if Γfin dominates
the total width, and vice versa. If Γinit and Γfin are the dominant partial widths and they
are comparable to each other, then the strength is even higher.
As discussed in Appendix A.1, our narrow resonance rate in eq. (2.13) improves on
the form of the usual expression for narrow resonance in two ways: (a) it extends to the
subthresold domain; and (b) it introduces the factor f which accounts for a finite Eres/Γtot
ratio.
It is important to make a systematic and comprehensive search for all possible experimen-
tally identified resonances capable of removing this discrepancy. In addition, it is possible
that resonances and indeed energy levels themselves were missed, especially at the higher
energies, where uncertainties are greater. Therefore it is useful to map the parameter space
where the lithium discrepancy is removed to apriori lay down our expectations of such missed
resonances. This can be done by looking at interesting initial states involving 7Li and 7Be,
and abundant projectiles p, n, d, t, 3He, α, and parametrizing the effect of inelastic channels
on the mass-7 abundance. This is described in § 2.4.
2.4 Narrow resonance solution space
In order to study the effect of resonances in different compound nuclei on the abundance of
mass 7, our strategy is to specify the reaction rate for possible resonances, and then run the
BBN code to find the mass-7 abundance in the presence of these resonances. In particular,
for reactions involving light projectile X, we are interested in considering the general effect
28
of states 7A+X → C∗, including those associated with known energy levels in the compound
nucleus, as well as possible overlooked states.
We assume that the narrow resonance approximation holds true at least as a rough guide.
If the reaction pathway is specified, i.e., all of the nuclei 7A+X → C∗ → Y +Z are identified,
then the reduced mass µ, reverse ratio and the Q-value are uniquely determined. In this
case, the thermally averaged cross-section is given by eq. (2.13), with two free parameters:
the product ωΓeff and the resonance energy, Eres. Because the state C∗ is unspecified, so is
its spin J∗. On the other hand, we do know the spins of the initial state particles, and thus
ω is specified up to a factor 2J∗ +1 (eq. 2.10). For this reason, the ωΓeff dependence reduces
to (2J∗ + 1)Γeff , which we explicitly indicate in all of our plots.
In a few cases we will be interested in one specific final quantum state, e.g., 7Be(t, 3He)7Li;
when the final state is specified, the reaction can be completely determined, including the
effect of the reverse rates. However, in most situations we are interested in the possibility
of an overlooked excited state in the compound nucleus, and thus in unknown final states.
In this scenario we thus have only a “generic” inelastic exit channel. Consequently, for such
plots we cannot evaluate the reverse reaction rate (which is in all interesting cases small)
and so we set the reverse ratio to zero.
The resonant rates are included in the BBN code, individually for compound nuclei with
an interesting initial state. The plots below show contours of constant, reduced mass-7
abundances. A general feature of all the plots, is the near linear relation between log Γeff
and Eres in the region of larger, positive values of Γeff and Eres. This can seen quantitatively
as follows. The thermal rate is integrated over time or equivalently temperature to give the
final abundance of mass 7 or 7Li as it exists. Now assuming that the thermal rate operates at
an effective temperature, TLi, at which 7Li production peaks, a given value for this effective
〈σv〉 will give a fixed abundance. This implies,
δY7/Y7 ∼ 〈σv〉peak ∼ Γeff e−Eres/TLi ∼ constant (2.15)
29
This gives a feel for the linear relation in the plot.
2.4.1 A = 8 Compound Nucleus
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
8Li, 3+, 2.255 MeV 7Li + n 1 1 222.71 keV 33 ± 6 keV γ(ground state) 7.0 ± 3.0 × 10−2 eV(Included) 1 n (elastic) ≈100% 33 ± 6 keV
8Be, 2+, 16.922 MeV 7Li + p 1 2 -333.1 keV 74.0 ± 0.4 keV γ(ground state) 8.4 ± 1.4 × 10−2 eV1 γ(3.04 MeV) < 2.80 ± 0.18 eV2 α ≈ 100% ≈ 74.0 keV1 p (elastic) unknown
8Be, 1+, 17.640 MeV 7Li + p 1 1 384.9 keV 10.7 keV γ(ground state) 16.7 eV1 γ(3.04 MeV) 6.7 ± 1.3 eV2 γ(3.04 MeV) 0.12 ± 0.05 eV1 γ(16.63 MeV) (3.2 ± 0.3) × 10−2 eV1 γ(16.92 MeV) (1.3 ± 0.3) × 10−3 eV1 p (elastic) 98.8% 10.57 keV
8Be, 2−, 18.91 MeV 7Be + n 0 1 10.3 keV 122 keV* γ(16.922 MeV) 9.9 ± 4.3 × 10−2 eV(Included) 1 γ (16.626 MeV) 0.17 ± 0.07 eV
0 p < 105.1 keV*2 p+ 7Li
∗(0.4776 MeV) < 105.1 keV*
0 n (elastic) 16.65 keV*8Be, 3+, 19.07 MeV 7Be + n 1 1 170.3 keV 270 ± 20 keV p ≈ 100% < 270 keV
(Included) 3 p+ 7Li∗
(0.4776 MeV) < 270 keV1 γ(3.03 MeV) 10.5 eV1 n (elastic) unknown
8Be, 3+, 19.235 MeV 7Be + n 1 1 335.3 keV 227 ± 16 keV p ≈ 50% ≈ 113.5 keV(Included) 1 γ(3.03 MeV) 10.5 eV
1 n (elastic) ≈ 50% ≈ 113.5 keV8Be, 1−, 19.40 MeV 7Be + n 0 0 500.3 keV 645 keV p unknown
0 p+ 7Li∗
(0.4776 MeV) unknown0 n (elastic) unknown1 α unknown
8Bg.s.
, 2+, 0 MeV 7Be + p 1 1 -0.1375 MeV unknown p (elastic) unknown0 EC→ 8Be 8.5 × 10−19 eV
8B, 1+, 0.7695 MeV 7Be + p 1 1 630 ±3 keV 35.7 ± 0.6 keV γ (ground state) 25.2 ± 1.1 meV(Included) 1 p (elastic) 100% 35.7 ± 0.6 keV
Table 2.1: This table lists the potential resonances in 8Li, 8Be and 8B which may achieverequired destruction of mass 7. These are all allowed by selection rules and includes someresonances already accounted for in determining the current theoretical 7Li abundance in-dicated as (Included). The entrance and exit channels along with their partial and totalwidths (Γtot), minimum angular momenta (Linit, Lfin) as well as resonance energies are listedwherever experimental data are available. The starred widths are a result of fits from R-matrix analysis. The list includes final products in ground and excited states with the lattermarked with a star in the superscript.
As seen in Table 2.1, the only resonance energy level of interest in the 8Li compound
nucleus at 2.255 MeV is already accounted for in the 7Li+n reaction. In the 8Be compound
nucleus, there are six levels of relevance for destroying either 7Li or 7Be at 16.922, 17.64,
18.91, 19.07, 19.24 and 19.40 MeV within our limit on Eres. The 16.922 MeV level is more
30
than 300 keV below threshold and has a maximum total width of only 74 keV. Therefore,
it is expected to have a weak effect. The 17.64 MeV level has typically low photon widths
(≈ 20 eV ) and a total width of 10.7 keV. But this state’s decay is dominated by the elastic
channel which makes this channel uninteresting.
The energy level diagram for 8Be 7(Tilley et al., 2004) shows the initial state, 7Be + n at
an entrance energy of E = 18.8997 MeV bringing the 18.91, 19.07, 19.235 and 19.40 MeV
levels into play. From among these the effect of the 18.91, 19.07 and 19.235 MeV resonances
are already accounted for in the well known 7Be(n, p)7Li reaction Cyburt (2004a). The
(18.91 MeV, 2−), resonance with Linit = 0 is the dominant contributor Coc et al. (2004b);
Adahchour & Descouvemont (2003). Being a broad resonance with a total width of ≈ 122
keV, the Breit-Wigner form is not used and instead an R-matrix fit to the data Cyburt
(2004a), is used to evaluate the contribution of the resonant rate. The remaining level
at 19.40 should also contribute to this reaction through ground and excited states. Only
the 19.40 MeV channel can have an α exit channel due to parity considerations. And this
resonance, despite a high resonance energy of ≈ 500 keV, can in principle be important due
to its large total width of 645 keV, if the proton branching ratio is high.
Figure 2.1 shows the 7Li abundance in the (Γeff , Eres) plane for the 7Be(n, p)7Li reaction.
Contours for 7Li/H ×1010 = 1.23, 2.0, 3.0, 4.0, and 5.0 (as labelled) are plotted as functions
of the effective width and resonant energy. Below ≈ (2J∗ + 1)40 = 120 keV, we expect
our results based on the narrow resonance approximation to be quite accurate. As one can
see from this figure, to bring the 7Li abundance down close to observed values, one would
require a very low resonance energy (of order ±30 keV) with a relatively large effective width.
Unfortunately, the 19.40 MeV level of 8Be corresponds to Eres = 500 keV as shown by the
vertical dashed line and does not make any real impact on the 7Li abundance.
Figure 2.2 shows the effect of a 8B resonance with 7Be and p in the initial state, plotted
in the (Γeff , Eres) plane with contours of constant mass 7 abundances. According to Fig. 2.2
7http : //www.tunl.duke.edu/nucldata/figures/08figs/08 04 2004.gif.
31
Figure 2.1: The effect of resonances in the 8Be compound nucleus involving initial states7Be+n. It shows the range of values for the product of the resonant state spin degeneracyand resonance strength, (2J + 1)Γeff , versus the resonance energy. Contours indicate wherethe lithium abundance is reduced to 7Li/H = 1.23×10−10 (red), 2.0×10−10 (blue), 3.0×10−10
(green) 4.0 × 10−10 (black) and 5.0 × 10−10 (magenta). Normal resonances have Eres > 0,while subthreshold resonances lie in the Eres < 0. The horizontal dot-dashed black line is theexperimental value of the strength of the resonance corresponding to the 19.40 MeV energylevel. The vertical dashed black line shows the position of Eres for the same state.
32
for resonance energies of a few 10’s of keV’s, resonance strength of a few meV is sufficient
to attain the observational value of mass 7. However, from the energy level diagram for 8B,
8(Tilley et al., 2004), the closest resonant energy level, Eex is at 0.7695 MeV 9(Tilley et al.,
2004), whose effect is already included via the 7Be(p, γ)8B reaction. The experimental value
of resonance energy is 632 keV which is off the scale in this figure. The only other close
energy level to the 7Be+ p entrance channel is at -0.1375 MeV which means that the ground
state is a sub-threshold state. This is not the usual resonant reaction, since the ground state
doesn’t have a width in the sense we refer to a width for the other reactions. But at these
energies, the astrophysical S-factor is ≈ 10 eV-barn which is very small and will yield a low
cross-section. This too is off scale in the figure and verifies that the 7Be(p, γ)8B reaction
does not yield an important destruction channel.
2.4.2 A = 9 Compound Nucleus
The energy level diagram for 9Be, 10(Tilley et al., 2004) shows energy levels of interest at
16.671, 16.9752 and 17.298 MeV; these appear in Table 2.2. The 7Li + d entrance channel
sits at 16.6959 MeV. The lowest lying resonant state is at 16.671 MeV and is a sub-threshold
state with Eres = −24.9 keV which lies within the total width of 41 keV. This resonance
is thus obviously tantalizing–it is well-tuned energetically and involves an abundant, stable
projectile. The 7Li abundance contours for the 9Be resonance states are shown in Fig. 2.3.
Perhaps disappointingly, the figure shows that the effect on primordial mass 7 is minor. This
illustrates the inability of direct 7Li destruction channels to reduce the mass-7 abundance,
as explained in §2.2. Given the overall difficulty of this channel, it is clear that the other
possible resonant energy levels (16.9752 MeV and 17.298 MeV) also fail to substantially
reduce the mass-7 abundance.
The 9B compound nucleus is relevant for studying the effect of the 7Be(d, p)2α and its
8http : //www.tunl.duke.edu/nucldata/figures/08figs/08 05 2004.gif.9http : //www.tunl.duke.edu/nucldata/figures/08figs/08 05 2004.gif.
10http : //www.tunl.duke.edu/nucldata/figures/09figs/09 04 2004.gif.
33
Figure 2.2: As in Fig. 2.1 for the resonances in the 8B compound nucleus involving initialstates 7Be+p.
34
Figure 2.3: As in Fig. 2.1 for the resonances in the 9Be compound nucleus.
35
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
9Be, 7Li + d 1 unknown -24.9 keV 41 ± 4 keV γ unknown(5/2+), 16.671 MeV 2 n+ 8Be unknown
0 n+ 8Be∗
(3.03 MeV) unknown2 n+ 8Be
∗(11.35 MeV) unknown
0 p unknown1 α unknown1 d (elastic) unknown
9Be, 7Li + d 0 1 279.3 keV 389 ± 10 eV γ (ground state) 16.9 ± 1.0 eV1/2−, 16.9752 MeV 1 γ (1.68 MeV) 1.99 ± 0.15 eV
2 γ (2.43 MeV) 0.56 ± 0.12 eV1 γ (2.78 MeV) 2.2 ± 0.7 eV
unknown γ (Unknown level TUNL) < 0.8 eV1 γ (4.70 MeV) 2.2 ± 0.3 eV1 p 12+12
−6 eV1 n < 288 eV1 n+ 8Be
∗(3.03 MeV) < 288 eV
3 n+ 8Be∗
(11.35 MeV) < 288 eV2 α < 241 eV0 d (elastic) 62 ± 10 eV
9Be, 7Li + d 0 unknown 602.1 keV 200 keV γ (ground state) unknown(5/2)−, 17.298 MeV 1 p 194.4 keV
(Included) 3 n+ 8Be unknown1 n+ 8Be
∗(3.03 MeV) unknown
1 n+ 8Be∗
(11.35 MeV) unknown2 α unknown0 d (elastic) unknown
Table 2.2: As in Table 2.1 listing the potential resonances in 9Be.
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
9B, (5/2+), 16.71 MeV 7Be + d 1 2 219.9 keV unknown p+ 8Be unknown0 p+ 8Be
∗(3.03 MeV) unknown
2 p+ 8Be∗
(11.35 MeV) unknown0 p+ 8Be
∗(16.626 MeV) unknown
0 p+ 8Be∗
(16.922 MeV) unknown2 3He unknown1 α + 5Li unknown3 α + 5Li
∗(1.49 MeV) unknown
1 d (elastic) unknown9B, (1/2−), 17.076 MeV 7Be + d 0 1 585.9 keV 22 keV p+ 8Be unknown
1 p+ 8Be∗
(3.03 MeV) unknown3 p+ 8Be
∗(11.35 MeV) unknown
1 p+ 8Be∗
(16.626 MeV) unknown1 3He unknown2 α + 5Li unknown0 α + 5Li
∗(1.49 MeV) unknown
0 d (elastic) unknown
Table 2.3: As in Table 2.1 listing the potential resonances in 9B.
competitors such as 7Be(d, 3He)6Li and 7Be(d, α)5Li. As seen in Table 2.3, the only two
levels of interest here are the 16.71 and 17.076 MeV levels. The 16.71 MeV level corresponds
to a resonance energy of 220 keV as shown by the vertical dashed line in Fig. 2.4 11(Tilley
et al., 2004). The widths are unknown experimentally. The approximate narrow resonance
11http : //www.tunl.duke.edu/nucldata/figures/09figs/09 05 2004.gif.
36
limit on the resonance width which is shown by the horizontal solid line is around 40 keV.
The p exit channel leads to the 7Be(d, p)8Be∗
reaction through the excited state at 16.63
MeV in 8Be. This should eventually lead to formation of alpha particles. Fig. 2.4 shows the
effect of the 16.71 MeV resonance on the mass-7 abundance as a function of the resonance
strength and energy under the narrow resonance approximation. From the plot, we see that
the 7Li abundance is reduced by 50% for (2J + 1)Γeff = 240 keV. This state has J = 5/2
and therefore, a value Γeff = 40 keV or more will have substantial impact on the problem.
Furthermore, as ΓL ≥ Γeff , we require ΓL >∼ 40 keV. This result confirms the conclusion of
Cyburt & Pospelov (2012). Later in § 2.5 we will see how this compares with theoretical
limits. As the decay widths are largely unknown, experimental data on the width are needed.
The state at 17.076 MeV corresponds to a resonant energy of Eres = 586 keV and is
beyond the scale shown in Fig. 2.4. A solution using this state is very unlikely.
2.4.3 A = 10 Compound Nucleus
Table 2.4 shows that the 10Be compound nucleus has energy levels at 17.12 and 17.79 MeV
12(Tilley et al., 2004) which are close to the initial state 7Li + t at 17.2509 MeV. The former
is far below threshold and does not contribute to 7Li destruction. The 17.79 MeV level is
around 540 keV above the entrance energy and its spin and parity are unknown. The total
width 13(Tilley et al., 2004) is Γtot = 112 keV which implies a small overlap with the entry
channel which renders this resonance insignificant despite having a number of n exit channels
with both ground state and excited states of 9Be. As seen in Fig. 2.5, the effect of 7Li + t is
small for the interesting region of parameter space.
The 10B compound nucleus has energy levels at 18.2, 18.43, 18.80 and 19.29 MeV, which
we investigate. The 18.2 MeV level is uncertain experimentally 14(Tilley et al., 2004) as
indicated in Table 2.5, and hence ideal for parametrizing. There is a 3He entrance channel a
12http : //www.tunl.duke.edu/nucldata/figures/10figs/10 04 2004.gif.13http : //www.tunl.duke.edu/nucldata/figures/10figs/10 04 2004.gif.14http : //www.tunl.duke.edu/nucldata/figures/10figs/10 05 2004.gif.
37
Figure 2.4: As in Fig. 2.1 for the resonances in the 9B compound nucleus. The verticaldashed line at 220 keV indicates the experimental central value of the resonance energy ofthe 16.71 MeV level.
38
Compound Nucleus, Initial State Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex Channels Width10Be, 7Li + t 0 0 -130.9 keV ≈ 150 keV n+ 9Be unknown
(2−), 17.12 MeV 1 n+ 9Be∗
(1.684 MeV) unknown0 n + 9Be
∗(2.4294 MeV) unknown
2 n+ 9Be∗
(2.78 MeV) unknown1 n+ 9Be
∗(3.049 MeV) unknown
1 n+ 9Be∗
(4.704 MeV) unknown0 n+ 9Be
∗(5.59 MeV) unknown
2 n+ 9Be∗
(6.38 MeV) unknown3 n+ 9Be
∗(6.76 MeV) unknown
0 n+ 9Be∗
(7.94 MeV) unknown0 t (elastic) unknown
10Be, 7Li + t unknown unknown 539.1 keV 112 ± 35 keV γ 3 + 2 eVunknown, 17.79 MeV unknown n+ 9Be < 77 keV
unknown n+ 9Be∗
(1.684 MeV) < 77 keVunknown n + 9Be
∗(2.4294 MeV) < 77 keV
unknown n+ 9Be∗
(2.78 MeV) < 77 keVunknown n+ 9Be
∗(3.049 MeV) < 77 keV
unknown n+ 9Be∗
(4.704 MeV) < 77 keVunknown n+ 9Be
∗(5.59 MeV) < 77 keV
unknown n+ 9Be∗
(6.38 MeV) < 77 keVunknown n+ 9Be
∗(6.76 MeV) < 77 keV
unknown n+ 9Be∗
(7.94 MeV) < 77 keVunknown t (elastic) 78 keV
Table 2.4: As in Table 2.1 listing the potential resonant reactions in 10Be.
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
10B, 7Li + 3He unknown unknown 411.7 keV 1500 ± 300 keV p+ 9Be unknownunknown, (18.2 MeV) unknown p+ 9Be
∗(1.684 MeV) unknown
unknown p+ 9Be∗
(2.4294 MeV) unknownunknown p+ 9Be
∗(2.78 MeV) unknown
unknown p+ 9Be∗
(3.049 MeV) unknownunknown p+ 9Be
∗(4.704 MeV) unknown
unknown p+ 9Be∗
(5.59 MeV) unknownunknown p+ 9Be
∗(6.38 MeV) unknown
unknown p+ 9Be∗
(6.76 MeV) unknownunknown p+ 9Be
∗(7.94 MeV) unknown
unknown p+ 9Be∗
(11.283 MeV) unknownunknown n+ 9B unknownunknown n + 9B
∗(1.6 MeV) unknown
unknown n + 9B∗
(2.361 MeV) unknownunknown n+ 9B
∗(2.75 MeV) unknown
unknown n + 9B∗
(2.788 MeV) unknownunknown n + 9B
∗(4.3 MeV) unknown
unknown n+ 9B∗
(6.97 MeV) unknownunknown d+ 8Be unknownunknown d+ 8Be
∗(3.03 MeV) unknown
unknown d+ 8Be∗
(11.35 MeV) unknownunknown t unknownunknown α + 6Li unknownunknown α+ 6Li
∗(2.186 MeV) unknown
unknown α+ 6Li∗
(3.563 MeV) unknownunknown α + 6Li
∗(4.31 MeV) unknown
unknown α + 6Li∗
(5.37 MeV) unknownunknown 3He (elastic) unknown
Table 2.5: As in Table 2.1 listing the ground and excited final states for the 18.2 MeV energylevel in 10B.
39
Figure 2.5: As in Fig. 2.1 for the resonances in the 10Be compound nucleus involving theinitial state 7Li+t (left), and in the 10B compound nucleus involving the initial state 7Li+3He.(right).
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
10B, 7Li + 3He 0 unknown 641.7 keV 340 keV γ(ground state) ≥ 3 eV2−, 18.43 MeV unknown γ (4.77 MeV) ≥ 17eV
0 n+ 9B unknownunknown n + 9B
∗(1.6 MeV) unknown
0 n+ 9B∗
(2.361 MeV) unknown2 n+ 9B
∗(2.75 MeV) unknown
1 n+ 9B∗
(2.788 MeV) unknownunknown n + 9B
∗(4.3 MeV) unknown
2 n+ 9B∗
(6.97 MeV) unknown0 p + 9Be unknown1 p+ 9Be
∗(1.684 MeV) unknown
0 p+ 9Be∗
(2.4294 MeV) unknown2 p+ 9Be
∗(2.78 MeV) unknown
1 p+ 9Be∗
(3.049 MeV) unknown1 p+ 9Be
∗(4.704 MeV) unknown
0 p+ 9Be∗
(5.59 MeV) unknown2 p+ 9Be
∗(6.38 MeV) unknown
3 p+ 9Be∗
(6.76 MeV) unknown0 p+ 9Be
∗(7.94 MeV) unknown
2 p+ 9Be∗
(11.283 MeV) unknown0 p+ 9Be
∗(11.81 MeV) unknown
1 d+ 8Be unknown1 d+ 8Be
∗(3.03 MeV) unknown
1 d+ 8Be∗
(11.35 MeV) unknown1 α + 6Li unknown1 α + 6Li
∗(2.186 MeV) unknown
1 α + 6Li∗
(4.31 MeV) unknown1 α + 6Li
∗(5.37 MeV) unknown
1 α + 6Li∗
(5.65 MeV) unknown0 3He (elastic) unknown
Table 2.6: As in Table 2.1 listing the ground and excited final state channels for the 18.43MeV energy level in 10B for the 7Li + 3He initial state.
little over 400 keV below this level. The current total experimental width is 1.5 MeV which is
very large and the branching ratios are unknown. The current uncertainty in the Eres is 200
40
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
10B, 7Be + t 0 unknown -239.1 keV 340 keV γ(ground state) ≥ 3 eV2−, 18.43 MeV unknown γ (4.77 MeV) ≥ 17 eV
0 n + 9B unknownunknown n+ 9B
∗(1.6 MeV) unknown
0 n+ 9B∗
(2.361 MeV) unknown2 n + 9B
∗(2.75 MeV) unknown
1 n+ 9B∗
(2.788 MeV) unknownunknown n+ 9B
∗(4.3 MeV) unknown
2 n + 9B∗
(6.97 MeV) unknown0 p + 9Be unknown1 p+ 9Be
∗(1.684 MeV) unknown
0 p+ 9Be∗
(2.4294 MeV) unknown2 p+ 9Be
∗(2.78 MeV) unknown
1 p+ 9Be∗
(3.049 MeV) unknown1 p+ 9Be
∗(4.704 MeV) unknown
0 p+ 9Be∗
(5.59 MeV) unknown2 p+ 9Be
∗(6.38 MeV) unknown
3 p+ 9Be∗
(6.76 MeV) unknown0 p+ 9Be
∗(7.94 MeV) unknown
2 p+ 9Be∗
(11.283 MeV) unknown0 p+ 9Be
∗(11.81 MeV) unknown
1 d+ 8Be unknown1 d+ 8Be
∗(3.03 MeV) unknown
1 d+ 8Be∗
(11.35 MeV) unknown1 α+ 6Li unknown1 α + 6Li
∗( 2.186 MeV) unknown
1 α + 6Li∗
( 4.31 MeV) unknown1 α + 6Li
∗( 5.37 MeV) unknown
1 α + 6Li∗
( 5.65 MeV) unknown0 t (elastic) unknown
Table 2.7: As in Table 2.1 listing the ground and excited final state channels for the 18.43MeV energy level in 10B for the 7Be + t initial state.
keV. However, according to the plot in Fig. 2.5, even a 200 keV reduction in Eres would not
be sufficient to cause any appreciable destruction of 7Li as this reaction has negligible effect
on the mass-7 abundance. This is another illustration of the fact that reactions involving
direct destruction of 7Li are unimportant.
The 18.43 MeV level is better understood Yan et al. (2002) and with a resonance energy
of ∼ 640 keV for the 7Li + 3He initial state (Table 2.6) and a total width of 340 keV has
a lower entrance probability and therefore is likely to be ineffective. This is evident from
Fig. 2.5. This level is also a sub-threshold resonance for the 7Be + t state (Table 2.7), with
resonance energy, Eres = −239.1 keV. This is far below threshold rendering it ineffective.
Staying with 7Be+ t, the closest energy level above the entrance energy of 18.669 MeV is
the 18.80 MeV (2+) level (Table 2.8), which corresponds to a resonance energy of ≈ 130 keV
41
Compound Nucleus, Initial State Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex Channels Width
10B, 7Be + t 1 unknown 130.9 keV < 600 keV γ (0.72 MeV) ≥ 20 eV2+, 18.80 MeV unknown γ (3.59 MeV) ≥ 20 eV
1 n + 9B unknownunknown n+ 9B
∗(1.6 MeV) unknown
1 n + 9B∗
(2.361 MeV) unknown1 n+ 9B
∗(2.75 MeV) unknown
0 n + 9B∗
(2.788 MeV) unknownunknown n+ 9B
∗(4.3 MeV) unknown
1 n+ 9B∗
(6.97 MeV) unknown1 p + 9Be unknown2 p+ 9Be
∗(1.684 MeV) unknown
1 p+ 9Be∗
(2.4294 MeV) unknown1 p+ 9Be
∗(2.78 MeV) unknown
0 p+ 9Be∗
(3.049 MeV) unknown0 p+ 9Be
∗(4.704 MeV) unknown
1 p+ 9Be∗
(5.59 MeV) unknown1 p+ 9Be
∗(6.38 MeV) unknown
2 p+ 9Be∗
(6.76 MeV) unknown1 p+ 9Be
∗(7.94 MeV) unknown
1 p+ 9Be∗
(11.283 MeV) unknown1 p+ 9Be
∗(11.81 MeV) unknown
2 d+ 8Be unknown0 d+ 8Be
∗(3.03 MeV) unknown
2 d+ 8Be∗
(11.35 MeV) unknown1 3He + 7Li unknown1 3He + 7Li
∗(0.47761 MeV) unknown
2 α + 6Li unknown2 α+ 6Li
∗(2.186 MeV) unknown
2 α + 6Li∗
(3.56 MeV) unknown0 α + 6Li
∗(4.31 MeV) unknown
0 α + 6Li∗
(5.37 MeV) unknown2 α + 6Li
∗(5.65 MeV) unknown
1 t (elastic) unknown
Table 2.8: As in Table 2.1 listing the ground and excited final state channels for the 18.80MeV energy level in 10B for the 7Be + t initial state.
15(Tilley et al., 2004). The exit channel widths for p and 3He are unknown experimentally and
thus, this is a candidate for parametrization. There is a weak upper limit on Γtot < 600 keV
16(Tilley et al., 2004), which for J∗ = 2 is off scale in Figs. 2.6 and 2.7. The contour plot
in Fig. 2.6 show that for a central value of resonance energy of ≈ 130 keV shown by the
vertical dashed line, resonance strength of just under 1 MeV is required which is very high.
Also, parity requirements force L = 1, which will cause suppression of this channel. We note
that there is no quoted uncertainty for this energy level and neighboring levels have typical
uncertainties of 100-200 keV. Therefore it may be possible (within 1-2σ) that the state lies
at an energy of 100 keV lower and would energetically, have a chance at solving the 7Li
15http : //www.tunl.duke.edu/nucldata/figures/10figs/10 05 2004.gif.16http : //www.tunl.duke.edu/nucldata/figures/10figs/10 05 2004.gif.
42
Compound Nucleus, Initial State Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex Channels Width
10B, 2−, 19.29 MeV 7Be + t 0 unknown 620.9 keV 190 ± 20 keV γ unknown0 n + 9B unknown
unknown n+ 9B∗
(1.6 MeV) unknown0 n + 9B
∗(2.361 MeV) unknown
2 n+ 9B∗
(2.75 MeV) unknown1 n + 9B
∗(2.788 MeV) unknown
unknown n+ 9B∗
(4.3 MeV) unknown0 p + 9Be unknown1 p+ 9Be
∗(1.684 MeV) unknown
0 p+ 9Be∗
(2.4294 MeV) unknown2 p+ 9Be
∗(2.78 MeV) unknown
1 p+ 9Be∗
(3.049 MeV) unknown1 p+ 9Be
∗(4.704 MeV) unknown
0 p+ 9Be∗
(5.59 MeV) unknown2 p+ 9Be
∗(6.38 MeV) unknown
3 p+ 9Be∗
(6.76 MeV) unknown0 p+ 9Be
∗(7.94 MeV) unknown
2 p+ 9Be∗
(11.283 MeV) unknown0 p+ 9Be
∗(11.81 MeV) unknown
1 d+ 8Be unknown1 d+ 8Be
∗(3.03 MeV) unknown
3 d+ 8Be∗
(11.35 MeV) unknown0 3He unknown1 α + 6Li unknown1 α+ 6Li
∗(2.186 MeV) unknown
1 α + 6Li∗
(4.31 MeV) unknown1 α + 6Li
∗(5.37 MeV) unknown
1 α + 6Li∗
(5.65 MeV) unknown0 t (elastic) unknown
Table 2.9: As in Table 2.1 listing the ground and excited final state channels for the 19.29MeV energy level in 10B for the 7Be + t initial state.
problem. This is true for the p exit channel.
The 3He exit channel may also reduce mass 7, through the formation of the 7Li which is
much easier to destroy. This is reflected in Fig. 2.7, which shows that at resonance energies
of ≤ 100 keV, a strength of a few 100 keV, but less than 600 keV may be sufficient to achieve
comparable destruction of 7Be as the 16.71 MeV resonance. The caveat is that for such values
of strengths, the narrow resonance approximation does not hold true and this may lead to a
reduced effect. Nevertheless, this is yet another case deserving a detailed comparison with
the theoretical limits which will follow in § 2.5. Once again, definitive conclusions can be
drawn only based on experimental data.
The 19.29 MeV level (Table 2.9) is energetically harder to access and with a total width
of only 190 keV, it is unlikely to be of significance, despite being less studied.
43
Figure 2.6: As in Fig. 2.1 for the resonances in the 10B compound nucleus involving initialstates 7Be+t.
44
Figure 2.7: As in Fig. 2.1 for the resonances in the reaction 7Be(t,3He)7Li.
45
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
10C, 7Be + 3He unknown unknown unknown unknown p unknownunknown unknown (Q = 15.003 MeV) α unknown
3He (elastic) unknown11B, 7Li + α 0 1 -103.7 keV 1.346 eV γ (ground state) 0.53 ± 0.05 eV
(3/2−), 8.56 MeV 1 γ (2.125 MeV) 0.28 ± 0.03 eV1 γ (4.445 MeV) (4.7 ± 1.1) × 10−2 eV1 γ (5.020 MeV) (8.5 ± 1.2) × 10−2 eV1 α (elastic) Unknown
11B, 7Li + α 2 1 256.3 keV 4.37 ± 0.02 eV γ (ground state) 4.10 ± 0.20 eV(5/2−), 8.92 MeV 2 γ (ground state) (5.0 ± 3.6) × 10−2 eV
(Included) 1 γ (4.445 MeV) 0.22 ± 0.02 eV1 α (elastic) Unknown
11B, 7Li + α 3 1 526.3 keV 1.9+1.5−1.1 eV γ (ground state) (2.7 ± 1.2) × 10−3eV
7/2+, 9.19 MeV 2 γ (4.445 MeV) 0.25 ± 0.09 eV0 γ (6.743 MeV) (3.8 ± 1.3) × 10−2 eV1 α (elastic) Unknown
11B, 7Li + α 1 1 606.3 keV 4 keV γ (ground state) 0.212 eV5/2+, 9.271 MeV 0 γ (4.445 MeV) 0.802 eV
0 γ (6.743 MeV) 0.137 eV1 γ (6.792 MeV) < 0.007 eV1 α (elastic) ≈ 4 keV
11C, 7Be + α 1 1 -43.3 keV 0.0105 eV γ (ground state) Unknown3/2+, 7.4997 MeV 0 γ (2.0 MeV) Unknown
1 α(elastic) Unknown11C, 7Be + α 0 1 557 keV 11 ± 7 eV γ (ground state) 0.26 ± 0.06 eV
(3/2−), 8.10 MeV 1 γ (2.0 MeV) (9.1 ± 2.3) × 10−2 eV0 α(elastic) Unknown
Table 2.10: As in Table 2.1 listing resonances in 10C, 11B and 11C.
The 10C nucleus 17(Ajzenberg-Selove, 1990) appearing in Table 2.10 shows large uncer-
tainties and experimental gaps at higher energy levels which may be relevant to entrance
channels involving 7Be. Reactions involving the 7Be + 3He initial state could contribute in
destroying 7Be if there exists a resonance in the parameter space shown in the Fig. 2.8.
These reactions win over those involving the 7Be+ t state, because 3He is substantially more
abundant than t, but are worse off due to a higher Coulomb barrier. The entrance energy
for 7Be + 3He is 15.0 MeV. As one can see from the figure, a 1− or 2− state with a resonance
energy of either -10 keV or 40 keV corresponding to energy levels of 14.99 and 15.04 MeV
respectively with a strength as high as a few 10’s of keVs is what it will take to solve the
lithium problem with this initial state. Thus, any 10C resonance near these energies which
may have been missed by experiment may be interesting as a solution to the lithium problem;
we return to this issue in more detail in § 2.5.
17http : //www.tunl.duke.edu/nucldata/figures/11figs/11 06 1990.gif.
46
Figure 2.8: As in Fig. 2.1 for the resonances in 10C involving initial state 7Be + 3He
2.4.4 A = 11 Compound Nucleus
For 11B 18(Ajzenberg-Selove, 1990), Table 2.10 shows that the entrance channel, 7Li+α is at
8.6637 MeV which is 103.7 keV above the resonant energy level at 8.560 MeV and ≈ 260 keV
below the resonant energy level at 8.92 MeV. Parity demands angular momentum to be 0.
Both states are at relatively large |Eres| and are not capable of making a sizable impact on
the 7Li abundance. Table 2.10 further lists states 9.19 MeV (which requires L = 3 and has
a total width of < 2 eV) and 9.271 MeV (whose decay is dominated by the elastic channel)
18http : //www.tunl.duke.edu/nucldata/figures/11figs/11 05 1990.gif.
47
which have progressively larger resonant energies and are unlikely to provide a solution.
For 11C 19(Ajzenberg-Selove, 1990), the entrance channel, 7Be+α is at 7.543 MeV which
is 43 keV above the resonant energy level at 8.560 MeV and 557 keV below the resonant
energy level at 8.10 MeV.
Figure 2.9: As in Fig. 2.1 for the resonances in 11C involving initial states 7Be+α.
As seen in Fig. 2.9, we find that the sub-threshold resonance in the 11C nucleus, produces
a very insignificant effect on 7Be in agreement with the claim in Cyburt & Pospelov (2012).
The super-threshold resonance states are also too far away at resonance energies, 557 keV
19http : //www.tunl.duke.edu/nucldata/figures/11figs/11 06 1990.gif.
48
and 260 keV for 7Be(α, γ)11C and 7Li(α, γ)11B respectively.
However, Fig. 2.9 shows that the presence of a (missed) resonance at resonance energies
of few tens of keV ’s, requires a very meagre strength of the order of tens of meV ’s to destroy
mass 7 substantially. Strengths of this order are typical of electromagnetic channels. It is
difficult to assess the probability that a 11C state at 7.55 MeV has been overlooked.
2.5 Reduced List of Candidate Resonances
Having systematically identified all possible known resonant energy levels which could affect
BBN, we find most of these levels are ruled out immediately as promising solutions, based on
their measured locations, strengths, and widths. As expected, the existing electromagnetic
channels are too weak to cause significant depletion of lithium owing to their small widths.
From amongst the various hadronic channels listed in the tables above, we have seen
that all channels are unimportant except three which are summarized in table 7.1. The
7Be+d channels involving the 16.71 MeV resonance in 9B, the 7Be+ t channels involving the
18.80 MeV resonance in 10B and 7Be + 3He channels. These are ones where a more detailed
theoretical calculation of widths is required to decide whether they may be important or not.
For each reaction, the Wigner limit, eq. (2.12), to the reduced width γ2 imposes a bound on
ΓL via eq. (2.11). Specifically, the penetration factor, PL(E, a), must be estimated to see if
the required strengths according to Figs. 2.4, 2.6 2.7, and 2.8 to solve the problem are at
all attainable. The penetration factor is given by
PL(E, a) =1
G2L(E, a) + F 2
L(E, a)(2.16)
where GL(E, a) and FL(E, a) are Coulomb wavefunctions.
We note that the Coulomb barrier penetration factor decreases as the energy of the
projectile and/or the channel radius, a, increases. For a narrow resonance, the relevant
projectile energy is E ≈ Eres, which is set by nuclear experiments (where available) and
49
their uncertainties. The channel radius corresponds to the boundary between the compound
nucleus in the resonant state and the outgoing / incoming particles. Therefore, the channel
radius depends on the properties of the compound state and the particles into which it
decays.
Consider the case of 7Be+d, which has resonance energy, Eres = 220±100 keV and initial
angular momentum, Linit = 1. A naive choice for the channel radius is the “hard-sphere”
approximation,
a12 = 1.45 (A1/31 + A
1/32 )fm (2.17)
which gives a27 = 4.6 fm. Using the Coulomb functions, Γ1 is of order a few keVs. The
corresponding strength, Γeff should be essentially the same and we further gain a factor of 6
from the spin of this state. This suggests by using figure 2.4, that this resonance should fall
short of the width required to solve or even ameliorate the problem.
However, reactions involving light nuclides including A = 7 are found to have channel
radii exceed the hard-sphere approximation Cyburt & Pospelov (2012). We thus consider
larger radii and find that for values higher than around 10 fm, we get a width which has the
potential to change the 7Li abundance noticeably. The Wigner limit
a2 =3~
2
2 µ Eres(2.18)
gives a larger radius, a27 = 13.5 fm, which gives one a better chance of solving the problem.
This is consistent with the conclusions drawn by Cyburt & Pospelov (2012).
For the 7Be + t initial state, the 18.80 MeV state of 10B has a resonance energy E =
0.131 MeV and Linit = 1. There is no experimental error bar on the resonance energy. The
hard sphere approximation gives a37 = 4.9 fm. This gives a width, Γ1, which is less than a
tenth of a keV, and is orders of magnitude lower than what is needed. In the spirit of what
we did in the earlier case, using eq. (2.18) gives a channel radius, a37 = 15 fm improving
50
the situation by almost 2 orders of magnitude in Γ1. If, in addition to increasing a37, the
resonance energy were to be higher by 100 keV, then Γ1 could be large enough to change
the 7Li abundance noticeably.
The 7Be+ 3He initial state will have 10C as the compound state. The structure of the 10C
nucleus is not well studied experimentally 20(Ajzenberg-Selove, 1990) nor theoretically. In
particular, we are unaware of any published data on 10C states near the 7Be + 3He entrance
energy, i.e., states at or near Eex(10C) ≈ Q(7Be + 3He) = 15.003 MeV. To our knowledge,
there has not been any search for narrow states in this region. The potential exit channels
of importance are 9B + p and 6Be + α. Because Jπ(3He) = 1/2+ Jπ(7Be) = 3/2−, to have
Linit = 0 and thus no entrance angular momentum barrier would require the 10C state to
have
Jπ = (1 or 2)− (2.19)
Because Jπ(9B) = 3/2−, the entrance channel spin and parity required to give Linit = 0 will
also allow Lfin = 0 for the 9B+p. On the other hand, in the final state 6Be+α both 6Be and
4He have Jπ = 0+. Thus if the putative 10C state has Jπ = 1−, this forces the 6Be + α final
state to have Lfin = 1, and thus this channel will be suppressed by an angular momentum
barrier relative to 9B + p.
Using eq. (2.17), we again get a37 = 4.9 fm. Taking Linit = 0 and E = 0.2 MeV, Γ0 is
about 10−3 keV and is extremely small. However, the penetration factor is highly sensitive
to the channel radius and a relatively small increase in a increases the width by orders of
magnitude. Increasing the energy does reduce the penetration barrier, but a higher width
is required due to thermal suppression. In order to get a sizable width, which is required to
solve the problem according to figure 2.8, a37must be >∼ 30 fm. At this energy, this radius
is somewhat larger than what is afforded by Eq. 2.18.
20http : //www.tunl.duke.edu/nucldata/figures/11figs/11 06 1990.gif.
51
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
9B, (5/2+), 16.71 MeV 7Be + d 1 0 219.9 keV unknown p+ 8Be∗
(16.63 MeV) unknown1 α + 5Li unknown
10B, 7Be + t 1 1 130.9 keV < 600 keV p+ 9Be∗
(11.81 MeV) unknown2+, 18.80 MeV 1 3He unknown
2 α unknown10C, 7Be + 3He unknown unknown unknown unknown p unknown
unknown unknown (Q = 15.003 MeV) α unknown3He (elastic) unknown
Table 2.11: This table lists surviving candidate resonances.
2.6 Discussion and Conclusions
The lithium problem was foreshadowed before precision cosmic microwave background data,
was cast in stark light by the first-year WMAP results, and has only worsened since. While
astrophysical solutions are not ruled out, they are increasingly constrained. Thus, a serious
and thorough evaluation of all possible nuclear physics aspects of primordial lithium pro-
duction is urgent in order to determine whether the lithium problem truly points to new
fundamental physics.
Reactions involving the primordial production of mass-7, and its lower-mass progenitor
nuclides, are very well studied experimentally and theoretically, and leave no room for sur-
prises at the level needed to solve the lithium problem Cyburt & Pospelov (2012); Boyd
et al. (2010b); Cyburt et al. (2003). Lithium destruction reactions are less well-determined.
While the dominant destruction channels 7Be(n, p)7Li and 7Li(p, α)α have been extensively
studied, in contrast, the subdominant destruction channels are less well-constrained.
We therefore have exhaustively cataloged possible resonant, mass-7 destruction channels.
As evidenced by the large size of Tables 2.1–2.10, the number of potentially interesting
compound states is quite large. However, it is evident that the basic conservation laws such
as angular momentum and parity coupled with the requirement of resonant reactions to be
2–3 times the 7Be(n, p)7Li rate prove to be extremely restrictive on the options for a resonant
solution to the lithium problem, and reduces the possibilities dramatically.
Given existing nuclear data, there are several choices for experimentally identified nuclear
52
resonances which come close to removing the discrepancy between the lithium WMAP+BBN
predictions and observations as tabulated in § 2.5. The 16.71 MeV level in 9B compound
nucleus, and the 18.80 MeV level in the 10B compound nucleus are two such candidates. It
is possible, however, that resonant effects have been neglected in reactions passing through
states which have been entirely missed. In all of the plots above, we have illustrated the
needed positions and strengths of such states, if they exist. One possibility involving the
compound state 10C is poorly studied experimentally, especially at higher energy states close
to the Q-value for 7Be + 3He.
Any of these resonances (or a combination) could offer a partial or complete solution
to the lithium problem, but in each case, we find that large channel radii (a > 10 fm) are
needed in order that the reaction widths are large enough. We confirm the results of Cyburt
and Pospelov Cyburt & Pospelov (2012) in this regard concerning 7Be + d, and we also find
similar channel radii are needed for 7Be + t, while larger radii are required for 7Be + 3He.
Obviously, nature need not be so kind (or mischievous!) in providing such fortuitious fine-
tuning. But given the alternative of new physics solutions to the lithium problem, it is
important that all conventional approaches be exhausted.
Thus, based on our analysis, quantum mechanics could allow resonant properties that
can remove or substantially reduce the lithium discrepancy. An experimental effort to mea-
sure the properties of these resonances, however can conclusively rule out these resonances
as solutions. If all possible resonances are measured and found to be unimportant for BBN,
this together with other recent work Boyd et al. (2010b), will remove any chance of a “nu-
clear solution” to the lithium problem, and substantially increase the possibility of a new
physics solution. Thus, regardless of the outcome, experimental probes of the states we have
highlighted will complete the firm empirical foundation of the nuclear physics of BBN and
will make a crucial contribution to our understanding of the early universe.
We are pleased to acknowledge useful and stimulating conversations with Robert Wiringa,
Livius Trache, Shalom Shlomo, Maxim Pospelov, Richard Cyburt, and Robert Charity. The
53
work of KAO was supported in part by DOE grant DE–FG02–94ER–40823 at the University
of Minnesota.
54
Chapter 3
Inverse-Compton Emission fromStar-forming Galaxies
Abstract
1 Fermi has resolved several star-forming galaxies, but the vast majority of the star-forming
universe is unresolved, and thus contributes to the extragalactic gamma ray background
(EGB). Here, we calculate the contribution from star-forming galaxies to the EGB in the
Fermi range from 100 MeV to 100 GeV, due to inverse-Compton (IC) scattering of the
interstellar photon field by cosmic-ray electrons. We first construct one-zone models for in-
dividual star-forming galaxies assuming supernovae power the acceleration of cosmic rays.
We develop templates for both normal and starburst galaxies, accounting for differences in
the cosmic-ray electron propagation and in the interstellar radiation fields. For both types of
star-forming galaxies, the same IC interactions leading to gamma rays also substantially con-
tribute to the energy loss of the high-energy cosmic-ray electrons. Consequently, a galaxy’s
IC emission is determined by the relative importance of IC losses in the cosmic-ray elec-
tron energy budget (“partial calorimetry”). We calculate the cosmological contribution of
star-forming galaxies to the EGB, using our templates and the cosmic star-formation rate
distribution. For all of our models, we find the IC EGB contribution is almost an order
of magnitude less than the peak of the emission due to cosmic-ray ion interactions (mostly
pionic pcrpism → π0 → γγ); even at the highest Fermi energies, IC is subdominant. The
flatter IC spectrum increases the high-energy signal of the pionic+IC sum, bringing it closer
to the EGB spectral index observed by Fermi. Partial calorimetry ensures that the overall
1This chapter matches the version which has been submitted to Astrophysical Journal, and posted versionon arxiv.org numbered on arXiv:1206.0770 and is co-authored with Brian Fields.
55
IC signal is relatively well-constrained, with only uncertainties in the amplitude and spectral
shape for plausible model choices. We conclude with a brief discussion on how the pionic
spectral feature and other methods can be used to measure the star-forming component of
the EGB.
3.1 Introduction
The window to the high-energy (> 30 MeV) gamma-ray cosmos has been open now for four
decades, with measurements of the diffuse emission by COS-B, (Caraveo et al., 1980; Mayer-
Hasselwander et al., 1980; Lebrun et al., 1982), OSO-3 satellite (Kraushaar et al., 1972)
followed by the second Small Astronomy Satellite (SAS-2)(Fichtel et al., 1975) and the En-
ergetic Gamma Ray Experiment Telescope (EGRET) (Hunter et al., 1997; Sreekumar et al.,
1998). SAS-2 was the first to reveal a diffuse, extra-Galactic gamma ray background (EGB).
More recently, the advent of the Fermi Gamma-Ray Space Telescope has substantially sharp-
ened our observational view of the EGB. With better energy and angular resolution and much
higher sensitivity than EGRET, Fermi has resolved many more gamma-ray point sources
and better determined the diffuse background and its energy dependence. The EGB data
are consistent with a power law of spectral index 2.41± 0.05 for energies > 100 MeV (Abdo
et al., 2010k).
The origin(s) of the EGB remains an open question. Contributions from active galaxies
(e.g., Strong et al., 1976a; Abdo et al., 2009; Stecker & Venters, 2011; Abazajian et al., 2011)
and star-forming galaxies (e.g., Strong et al., 1976b; Bignami et al., 1979; Pavlidou & Fields,
2002; Fields et al., 2010; Stecker & Venters, 2011; Makiya et al., 2011) are “guaranteed”
in the sense that these known, resolved extragalactic source classes must have unresolved
counterparts that will contribute to the EGB. Other possible EGB sources include truly
diffuse emission such as dark matter annihilation (Ando et al., 2007), interactions from
cosmic rays accelerated in structure formation shocks (e.g., Loeb & Waxman, 2000; Miniati,
56
2002), unresolved ordinary and millisecond pulsars (Faucher-Giguere & Loeb, 2010; Geringer-
Sameth & Koushiappas, 2012), and even Solar-System emission from cosmic-ray interactions
with Oort cloud bodies (Moskalenko & Porter, 2009).
One of Fermi’s major achievements has been to establish external star-forming galaxies
as a new class of gamma-ray sources. These detections give a global view of the gamma-ray
output as a result of star formation, complementary to the local gamma-ray images within
the Milky Way. Fermi has discovered γ-rays from “normal” galaxies undergoing quiescent
star formation, but only detected our very nearest neighbors, as anticipated (Pavlidou &
Fields, 2001). The Large Magellanic Cloud (LMC; Abdo et al., 2010i) has even spatially
resolved, and the SMC (Abdo et al., 2010d) and M31(Abdo et al., 2010f) have also been
detected. Other normal star-forming galaxies in the Local Group (including M33) have
not yet been seen (Abdo et al., 2010f; Lenain & Walter, 2011). Beyond the Local Group,
Fermi has detected starbursts galaxies characterised by very high star-formation rates, as
anticipated by Torres et al. (2004). Fermi has detected the starbursts M82 and NGC 253
(Abdo et al., 2010a) which had been detected at TeV energies by VERITAS (VERITAS
Collaboration et al., 2009) and HESS (Acero et al., 2009) respectively. In addition Fermi
has also detected the starbursts NGC 1068 and NGC 4945 (Nolan et al., 2012b).
The Fermi star-forming galaxies offer a qualitatively new probe of cosmic rays; they also
inform and calibrate efforts such as ours to understand the EGB contribution from the vast
bulk of the star-forming universe that remains unresolved. The LMC is the best resolved
individual system, and there the energy spectrum is consistent with pionic, while the spatial
distribution can be used to study cosmic-ray propagation (Murphy et al., 2012). More
broadly, the ensemble of all Fermi star-forming galaxies encodes information about global
cosmic-ray energetics and interaction mechanisms (Persic & Rephaeli, 2010; Lacki et al.,
2011; Persic & Rephaeli, 2012). In particular, Fermi reveals a strong correlation between
gamma-ray luminosity Lγ and supernova rate (or equivalently star-formation rate ψ). This
is expected if supernovae provide the engines of cosmic-ray acceleration. Remarkably, all
57
star-forming galaxies detected to date can be well-fit with a single power law Lγ ∝ ψ1.4±0.3
(e.g., Abdo et al., 2010f).
The main mechanism of gamma ray production in normal star-forming galaxies is an-
ticipated to be the same that dominates Milky Way diffuse gamma rays: pionic emission
pcrpism → ppπ0 → γγ, arising from interactions between cosmic-ray hadrons (ions) and inter-
stellar gas (Stecker & Venters, 2011; Abdo et al., 2009; Fields et al., 2010; Strong et al., 2010).
This mechanism is likely responsible for the non-linear relation between the luminosity of
Fermi galaxies and their star-formation rate. Namely, the observed correlation is consistent
with a picture (Pavlidou & Fields, 2001; Fields et al., 2010; Persic & Rephaeli, 2011) in
which the cosmic-ray proton flux is controlled by the supernova rate, the total number of
targets is set by the galaxy’s gas mass, and the gas and star-formation are linked by the
Schmidt-Kennicutt relation (Kennicutt, 1998).
Here, we examine the cosmological contribution of star-forming galaxies by the inverse-
Compton (IC) scattering ecrγis → eγ of cosmic-ray electrons on the interstellar radiation
field (ISRF). In order to do so, we construct a one-zone model of star-forming galaxies.
We normalize the IC emission (per unit star formation) in our model differently for normal
and for starburst galaxies. For normal galaxies we match to the Milky Way IC emission
as computed in GALPROP (Strong et al., 2010), while for starburst galaxies we set the
template matching M82 star formation rate and cosmic ray propagation consistent with
multiwavelength data just as other authors do (eg., Stecker & Venters, 2011; Lacki et al.,
2012; Paglione & Abrahams, 2012).
The IC gamma rays are produced by upscattering of interstellar radiation by high-energy
cosmic-ray electrons (Felten & Morrison, 1963; Felten, 1965; Brecher & Morrison, 1967).
Inverse-Compton scattering also represents an important energy loss mechanism for these
electrons; the other important losses are bremsstrahlung and synchrotron. The relative
importance of these losses depends on the cosmic-ray energy and on interstellar radiation
and matter densities. Where inverse Compton losses dominate, the energy injected into
58
cosmic-ray electrons is ultimately re-emitted as IC gamma-ray photons. This equality of
energy loss and energy output is known as calorimetry, and was first described in the context
of high-energy electrons by Felten (1965) and Voelk (1989) and explored in detail for diffuse
high-energy gamma rays by cosmic rays by Pohl (1994) and by Strong et al. (2010). If the
other losses are negligible then we have perfect calorimetry. But realistically, other losses
compete, and so the gamma-ray energy output is reduced by the IC fraction of the energy
losses by the cosmic rays.
In contrast to the loss-dominated propgations of cosmic-ray electrons, cosmic-ray hadrons
(mainly protons, as well as other ions) in the Milky Way suffer losses dominated by escape.
The pionic emission from normal star-forming galaxies is thus not calorimetric; however, in
the dense central regions of starburst galaxies, proton inelastic interactions can dominate
losses and lead to calorimetry (Lacki et al., 2010; H. E. S. S. Collaboration et al., 2012). In
the Milky Way, pionic emission dominates the total Galactic gamma-ray output (luminosity),
exceeding IC emission by factors of up to ∼ 5 in GALPROP calculations (Strong et al., 2010).
Our analysis will show that the inverse-Compton component from star-forming galaxies over
cosmological volumes is nearly an order of magnitude lower than the peak of the pionic
component.
The paper is organised as follows. Section 3.2 gives an order-of-magnitude estimate of the
inverse Compton EGB contribution by star-forming galaxies, and serves as an overview to our
paper. We first build a template for the IC emission from individual star-forming galaxies,
treating normal and starburst systems separately. We start with the various components of
the background interstellar photon field §3.3. These interstellar photons serve as scattering
targets for the cosmic ray electrons, whose propagation is explained in §3.4. For the highest
energy background photons and electrons, the scattering occurs in the Klein-Nishina regime,
which affects the emergent spectrum and is detailed in §3.5.1. Our one-zone model for the
inverse-Compton luminosity from a single galaxy is presented in §3.5.2. In §3.6, the total
intensity over cosmological volumes is calculated and compared with the pionic component.
59
Section 3.7 discusses the implications of our results.
3.2 Order-of-Magnitude Expectations
An order-of-magnitude estimate of our final result will help frame key physical issues and
astrophysical inputs. Our goal is to find the EGB contribution from inverse Compton scat-
tering in star-forming galaxies throughout the observable universe. We thus calcualte the
gamma-ray specific intensity, IE , at energy E. For photons up to at least <∼ 30 GeV, the
universe is optically thin;2 thus, the intensity is simply given by the line-of-sight integral
IE ≈ c
4π
∫
los
dℓ Lγ ≈ LγdH
4π(3.1)
where Lγ is the luminosity density or cosmic volume emissivity of the galaxies, and dH =
c/H0 = 3000h−1 Mpc is the Hubble length. The cosmic luminosity density of a distribution of
IC-emitting, star-forming galaxies can be expressed as a product of the luminosity function,
dngal/dLγ and the luminosity of each individual galaxy, Lγ:
Lγ =
∫
Lγdngal
dLγ
(Lγ) dLγ = 〈nLγ〉 ≈ ngalLE (3.2)
where LE is the luminosity of an average galaxy at energy E.
Within a single galaxy, the gamma-ray luminosity is an integral
Lγ(Eem) =
∫
qic dVe (3.3)
over the volume in which cosmic-ray electrons propagate. Here the inverse-Compton volume
emissivity qic depends on the product of the targets and projectile densities, and the inter-
action cross section. The density of targets is simply the number density of the interstellar
2We negelect the interactions with the extragalactic background light, leaing to pair production γγ →e+e−; these effects could be important above ∼ 100 GeV (e.g., Salamon & Stecker, 1998; Abdo et al., 2010e;Stecker et al., 2012).
60
photons, nisrf . The projectiles are cosmic-ray electrons, with flux density φe. The cross sec-
tion is that for inverse Compton scattering, dσic/dEem. Therefore, the emissivity for a single
galaxy is given as,
qic =
⟨
φedσic
dEem
nisrf
⟩
(3.4)
with the angle brackets indicating averaging over the cosmic-ray and background photon
energies.
We will assume that cosmic-ray acceleration is powered by supernova explosions. In-
deed, the cosmic-ray/supernova link historically has been better established for cosmic ray
electrons than for hadrons, thanks to radio (e.g., Webber et al., 1980; Strong et al., 2011;
Bringmann et al., 2012) and X-ray synchrotron (Uchiyama et al., 2007; Helder et al., 2009,
e.g.,) measurements. More recently Fermi-LAT (Volk et al., 1996; Ahlers et al., 2009) mea-
surements of pionic gamma rays provided the most direct evidence for supernova acceleration
of protons and other ions. Consequently, a galaxy’s cosmic-ray injection rate is proportional
to its supernova rate, qcr ∝ RSN. And because the supernova rate itself traces the galactic
star-formation rate ψ, we conclude that qcr ∝ ψ.
Cosmic-ray electron propagation is dominated by energy losses in the form of inverse
Compton, bremsstrahlung and synchrotron processes. Each contributes to a total energy
loss rate btot = |dEe/dt|. The IC loss rate is proportional to the background photon energy
density: bic ∝ Uisrf ∝ nisrf . The cosmic-ray flux φe is inversely proportional to the total energy
loss rate, btot φe ∝ (qe/ btot). Thus, IC emissivity, qic is lower than rate qe of energy injection
into cosmic-ray electrons, by the fraction bic/btot of energy losses in IC, qic ∝ (bic/ btot)qe. This
is the case of partial or fractional calorimetry where qic/qe ∝ bic/btot: IC photons trace the
portion of cosmic-ray energy lost via this mechanism (Pohl, 1994). Even the approximate
validity of calorimetry is sufficient that the IC gamma ray luminosity is a fairly robust
calculation.
61
Due to partial calorimetry, the IC volume emissivity is qic ∝ qe, and integration over all
of the supernovae acting as cosmic-ray accelerators gives the IC luminosity LE ∝ RSN ∝ ψ.
Physically, a fixed fraction of each supernova’s energy goes into cosmic-ray electrons and
ultimately into IC photons.
For normal star-forming galaxies, this scaling can then be compared to GALPROP esti-
mates for the total Milky-Way IC luminosity (Strong et al., 2010), which determines the IC
output per unit star-formation. We can then find the luminosity for any star-forming galaxy
at a fiducial energy of E = 1 GeV as,
E2 LE = E2 LE,MW
(
ψ
ψMW
)
≈ 1040 GeV2 sec−1 GeV−1
(
ψ
1 M⊙ yr−1
)
(3.5)
where ψMW is the Milky-Way star-formation rate.
From eqns. (3.1) (3.2), and (3.5), we estimate the IC contribution to the EGB intensity
at 1 GeV as
E2 IE∣
∣
ic,1GeV≈ LE,MW ngal dH
4 π
(
ψ
ψMW
)
≈ LE,MW dH
4 π ψMW
ρ⋆(1)
≈ 3 × 10−8 GeV cm2 sec−1 sr−1 (3.6)
where ρ⋆(1) = ψngal ≈ 0.1 M⊙ yr−1 Mpc−3 is the cosmic star formation rate, evaluated at
z = 1, near its peak (Horiuchi et al., 2009a). For comparison, the pionic model of Fields
et al. (2010) gives an intensity E2 IE|π→γγ,1GeV ∼ 3×10−7 GeV cm2 sec−1 sr−1, with a factor
∼ 2 uncertainty in normalisation. The EGB measured by Fermi-LAT is E2 IE|obs,1GeV = 6×
10−7 GeV cm2 sec−1 sr−1 (Abdo et al., 2010k). Thus, the pionic component alone dominates
the overall amplitude from the star-forming galaxies, which is an important contribution to
the total observed flux. Not evident from our estimate is that the shape of the star-forming
spectrum improves upon addition of the IC component, as we will see in §3.6.
62
3.3 Targets: Interstellar Photon Fields
Figure 3.1: The interstellar photon field (ISRF) energy density, shown as a function of thewavelength for normal (left panel) and starburst galaxies (right panel). (a) Left: Milky-Waytemplate for normal galaxies The model includes thermal components for starlight (optical),dust (IR) and the CMB as in Cirelli & Panci (2009). It includes another thermal componentfor the starlight in the UV. The black (fiducial), red and blue curves represent the curves for(R, h) = (0,5 kpc), (4 kpc,0) and (0,0) respectively. (b) Right: M82 template for starburstgalaxies. The solid black curve is a model (de Cea del Pozo et al., 2009) for a starburst ISRFdominated by IR emission due to the dusty nuclei. It has two thermal components, a coolone at 45 K and a warm one at 200 K as in de Cea del Pozo et al. (2009). These correspondto IR luminosity of 1010.5Lsol and a core radius of 200 pc. It is compared with the fiducialISRF of the Milky Way type normal galaxy shown in dashed black.
Cosmic-ray electrons in a galaxy will inverse-Compton upscatter ambient photons, which
are commonly called the interstellar radiation field (ISRF). Each galaxy’s ISRF is a rich
function of both energy and geometry. We simplify this complex reality by treating a galaxy
as a single zone for the purposes of cosmic-ray propagation and gamma-ray emission. Thus,
our ISRF is meant to give a sort of effective volume average of the radiation field encountered
by cosmic-ray electrons. A galaxy’s ISRF will evolve with redshift, and so for our single-
galaxy templates we must also create a prescription for how the different ISRF component
evolve with z.
63
Our choice of ISRF for normal galaxies follows those of models for the Milky Way, which
we adjust at other cosmic epochs via scaling arguments. These scalings use the cosmic
star-formation rate ρ⋆(z), whose change with redshift we characterize by the dimensionless
function
S(z) ≡ ρ⋆(z)
ρ⋆(0)(3.7)
where today S(0) = 1.
For the ISRF at the present day (redshift z = 0), we adopt but extend the model proposed
by Cirelli & Panci (2009), which consists of three thermal components corresponding to the
infra-red (or dust), optical, and the Cosmic Microwave Background (CMB). To this we add a
fourth thermal component for the UV. Our goal is to build a simple model that can reproduce
the inverse Compton luminosity of the Milky Way according to Strong et al. (2010); their
calculation is based on a far more detailed, spatially-dependent ISRF as implemented in the
GALPROP code (Strong et al., 2000; Porter et al., 2008). The strengths of the UV, IR and
the optical relative to the CMB decreases with Galactocentric radial distance R and also
with height h above the Galactic plane. This is seen in Fig. 3.1.
We may express the ISRF specific energy density duisrf/dǫ = ǫ dnisrf/dǫ as
λduisrf
dλ= ǫ
duisrf
dǫ=
4∑
i=1
fi1
π2
ǫ4
exp(ǫ/Ti) − 1(3.8)
a sum of Planck terms, with dimensionless weights fi for the different background compo-
nents: i = UV, optical, IR and the CMB. The UV and optical component are from starlight,
and the IR comes from starlight reprocessed by dust. Our model, as seen in the Fig. 3.1
thus includes the effect of the major components in the GALPROP ISRF, which dominate
the energy density. For the IR, we tried a non-thermal model that includes effect of both
scattering and absorption by dust particles, but the effects on our gamma ray luminosity
model are insignificant. Therefore, we retain the thermal model for IR. We also verified that
including the optical-IR transition where the non-thermal lines appear has negligible effect
64
on the final gamma ray luminosity.
Figure 3.1 shows the ISRF at different locations in the Galaxy. The optical and dust
components are taken from Cirelli & Panci (2009). We add another thermal component for
the UV. The black, red and blue curves correspond to the positions (R, h) = (0, 5), (4,0) and
(0,0) kpc. The peaks due to starlight and IR change with position in the Galaxy, while the
CMB contribution peaked at λ ≈ 1000 microns remains the same. At the Galactic centre, the
energy densities of these latter components are greater than the CMB, with optical photons
dominating. Above the Galactic plane, the CMB becomes comparable in energy density and
eventually dominates at very large h.
Table 3.1 lists the temperatures of the various ISRF components along with their relative
weights at zero redshift, fi(0) for the normal galaxies. We chose as a fiducial model the rela-
tive strengths for regions corresponding to 10 < |b| < 20 and 0 < l < 360, corresponding
to the ‘10-20’ model of Cirelli & Panci (2009). This is for the position (R, h) = (0, 5) Kpc
represented by the black curve in Fig. 3.1. The UV component at this position is scaled
from that in the Galactic Centre, (Porter et al., 2008) in the same proportion as the optical
component in these positions. This choice of model, with the overall normalisation adjusted,
reproduces the Milky Way IC luminosity as a function of energy from Strong et al. (2010).
The ISRF in starburst nuclei is dominated by the IR peak, which renders other compo-
nents relatively unimportant. Our one zone ISRF template is based on the two-component
model of de Cea del Pozo et al. (2009). The starburst ISRF is dominated by a thermal
peak corresponding to a temperature, T = 45 K, according to de Cea del Pozo et al. (2009).
The energy density is related to the IR luminosity, L from the nucleus of radius R as,
Uisrf = L/2πR2c (Lacki et al., 2010). Picking a normalisation based on the energy density
Uisrf ≈ 1100 eV cm−3, set by luminosity, L = 1010.5L⊙, and radius, R = 200 pc, we get the
green curve in the right panel of Fig. 3.1. However, according to de Cea del Pozo et al.
(2009), the dust emission fits better with an additional warm component with temperature
of 200 K. This is our fiducial model. The choice of the fiducial ISRF is displayed in solid
65
black in the right panel of Fig. 3.1. It is also summarised in the table 3.1. And we pick the
overall normalisations fi(0) in order match the same energy density as for the one component
case.
Table 3.1: ISRF Parameters for normal galaxiesMilky Way UV Optical IR CMB
Fiducial : fi(z = 0) 8.4 × 10−17 8.9 × 10−13 1.3 × 10−5 1Galactic Centre fi(z = 0) 1.6 × 10−15 1.7 × 10−11 7 × 10−5 1
T (z = 0) [K] 1.8 × 104 3.5 × 103 41 2.73
Table 3.2: ISRF Parameters for starburst galaxiesStarburst M82 type UV Optical IR (cool) IR (warm) CMBFiducial : fi(z = 0) 3.2 × 10−15 0.0 4.22 × 10−2 3.61 × 10−5 1
T (z = 0) [K] 1.8 × 104 3.5 × 103 45 200 2.73
Because we are interested in inverse Compton emission from star-forming galaxies over
all of cosmic history, we must specify the cosmic evolution of the ISRF. This will depend on
which stars contribute to starlight and dust scattered light. The detailed present-day Milky-
Way model in Strong et al. (2000) uses the stellar luminosity functions from Wainscoat et al.
(1992). The UV and IR components derive from short-lived massive stars, and the resulting
starlight density will be proportional to the star-formation rate. Thus, we adopt the scalings
f2(z) = fUV(z) = fUV(0) S(z),
f3(z) = fIR(z) = fIR(0) S(z) (3.9)
The optical component is less trivial and evolves differently for different stellar popula-
tions. Massive supernova progenitors, as well as AGB progenitors, are short-lived compared
to >∼ 1Gyr timescales for cosmic star formation. Consequently, the optical radiation density
of these stars would scale as
f2(z) = fop(z) = fop(0) S(z) , (3.10)
in step with the UV and IR scaling. For reasonable initial mass functions and star-formation
66
histories, these stars should dominate the optical ISRF. But lower-mass main sequence stars
and their red giant descendants have cosmologically long lifetimes; this means that their
starlight density at any epoch would scale as the integrated star-formation rate. Therefore,
their contribution to the optical component would scale as
f ′op(z) = fop(0)
∫∞
zdz| dt
dz| ρ⋆
∫∞
0dz| dt
dz| ρ⋆
, (3.11)
This would give a lower optical component and decrease the IC intensity as we will see. We
will adopt eq. (3.10) for our fiducial model, but we will explore the effects of using eq. (3.11);
these cases will bracket the true evolution.
The redshift dependence of the CMB is precisely known and can be expressed entirely in
terms of its temperature,
TCMB(z) = (1 + z) TCMB(0) (3.12)
with f4(0) = fCMB(0) = 1. The CMB energy density thus grows rapidly as Ucmb ∝ (1 + z)4.
3.4 Projectiles: Cosmic-Ray Electron Source and
Propagation
An electron with energy Ee = γmc2 scattering off a background photon of energy ǫ yields an
IC photo with energy Eγ ∼ γ2ǫ. In order to produce gamma rays around the Fermi range
Eγ ∼ 100 MeV to ∼ 300 GeV , electrons with energies of few GeV to few TeV are required,
depending on the energy of initial un-scattered photon. These are the electrons of interest
to us.
High-energy cosmic-ray electrons obey a complex transport equation (e.g., Strong &
Moskalenko, 1998). This in general would include effects of diffusion, convection, escape,
radiative losses, etc. However, our approach is to create the simplest one-zone model that
67
captures electron source spectrum and their most important losses.
We model cosmic-ray electron injection for normal galaxies via the source spectrum from
the Plain Diffusion (PD) model of Strong et al. (2010). In this model, electrons are accel-
erated with a broken power-law injection spectrum having qe(Ee) ∝ E−1.8e up to a break
at Ee,break = 4 GeV, and above this qe(Ee) ∝ E−2.25e . Electrons below Ee,break can produce
inverse Compton gamma rays at the low end of the Fermi range, when upscattering optical
and UV photons. Gamma rays at higher energies are produced by electrons above the break
energy. Thus, it is important to include the injection break in our calculations. Including
a cutoff at 2 TeV according to Strong et al. (2011), results in a rollover at high energies
improving the agreement with the GALPROP model beyond 100 GeV.
For starburst galaxies the injected cosmic-ray electrons important for IC emission are
difficult to constrain independently from gamma-ray emission, and thus are more uncertain.
For starburst galaxies, we use M82 as a template, and thus scale the injected spectrum up
from that of the Milky Way by the ratio ψM82/ψMW of star-formation rates, using ψM82 =
ψ8 M⊙ yr−1 in accordance with the total IR luminosity Sanders et al. (2003) as used by
Lacki et al. (2012). For our fiducial model we adpot an injection spectrum that is a single
power law qe(Ee) ∝ E−2.1e , although other indices ranging from 2.0 to 2.4 are permissible
(Paglione & Abrahams, 2012). The high-energy cutoff being less constrained (Lacki et al.,
2012) we pick fiducial value of 2 TeV, same as the Milky Way. Our fiducial model is directly
comparable with Paglione & Abrahams (2012).
Milky-Way cosmic-ray electrons experience an interstellar diffusion coefficient, D0 ∼
1028cm2sec−1 (Strong et al., 2010), so the diffusion length for electrons over the Compton
loss time-scales is ℓdiff ∼ 0.01 − 1 kpc, depending on the energy. We thus see that electrons
will not venture far from the Galaxy, but can range vertically outside the Galactic midplane.
Hence, very few electrons will be lost to escape, in strong contrast to ions for which escape
dominates (Pohl, 1994). We thus will ignore electron escape in our analysis.
However, since the electrons do propagate throughout and above the stellar and gaseous
68
disk, the electron population sees a wide variety of radiation fields. Our single choice is a
crude approximation meant to represent a sort of average ISRF. As a result, the electron
propagation equation takes the “thick target” form
∂
∂tφe(Ee) = qe(Ee) +
∂
∂Ee
[b(Ee)φe(Ee)] (3.13)
where qe(Ee) is the source term and b(Ee) = |dEe/dt| is the total rate of energy loss. The
equilibrium (i.e., ∂tφe = 0 steady state) solution of the cosmic ray flux is
φe(Ee) =1
b(Ee)
∫
Ee
dE ′e qe(E
′e) =
qe(> Ee)
b(Ee)(3.14)
As usual in the thick-target limit, the electron flux is directly proportional to the energy-
integrated injection rate qe(> Ee), and inversely proportional to the total energy loss rate.
In general, the total energy loss rate of the electrons is given by sum of the inverse Compton,
synchrotron, bremsstrahlung and ionisation losses,
btot(Ee) = bic(Ee) + bsync(Ee) + bbrem(Ee) ; (3.15)
we now address each in turn. The ionisation loss is unimportant throughout the entire
electron energy range of interest here.
The inverse Compton loss in the Thomson regime takes the simple form bic(Ee)Thoms
=
43σT c Uisrf
(
Ee
m c2
)2. However, the full Compton cross section takes the Klein-Nishina form
described in more detail in §3.5.1. Corrections to the Thomson limit become important
when ∆ǫ = 4 ǫγ/m c2 ≫ 1; in a starlight-dominated ISRF with ǫ ∼ 1 eV, this occurs for
electrons with γ ≫ 105. This energy regime is important for our calculation and therefore,
we calculate the inverse Compton loss rate by combining the Klein-Nishina cross section
69
self-consistently with the (redshift-dependent) ISRF:
bic(Ee) =
∫ ∞
0
dǫdnisrf
dǫ(ǫ)
∫
dEem (Eem − ǫ)dσic
dEem
= 3 σT c
∫ ∞
0
dǫdnisrf
dǫ(ǫ) ǫ
∫ 1
(m c2/2Ee)2dq
(4(
Ee
m c2
)2 − ∆ǫ)q − 1
(1 + ∆ǫq)3
×(
2q ln(q) + (1 + 2q)(1 − q) +1
2
(∆ǫq)2
1 + ∆ǫq(1 − q)
)
(3.16)
as in Cirelli & Panci (2009), where
q =Eem
4ǫγ2(1 − Eem/γm c2). (3.17)
Note also that for the highest-energy electrons, inverse Compton losses become catastrophic,
i.e., in each scattering event the fractional changes in the electron energy approach δEe/Ee ∼
1 (e.g., Blumenthal & Gould, 1970). In this case the losses can no longer be treated as a
smooth function bic(Ee) that averages over many scatterings each with δEe/Ee ≪ 1; we do
not include these effects, which impact gamma-ray energies of Eem >∼ 1 TeV.
The relative importance of the losses determines the propagated electron spectrum (Fel-
ten, 1965; Pohl, 1993). For our fiducial ISRF model, the bremsstrahlung, synchrotron and
Compton losses are all important, depending on the part of the spectrum. The rates are
given in the Appendix B.1. For electron energies lower than ∼ 1 GeV, bremsstrahlung is
dominant scaling as bbrem ∝ Ee, with Compton losses being comparable close to the break
energy at Ee = 4 GeV, scaling as bic ∝ E2e . Since, the electron injection spectral break and
the bremsstrahlung–Compton transition are fairly close, instead of seeing two breaks corre-
sponding to these two cosmic ray propagation regimes, only one is seen near the break energy
as described in §3.4. For large Ee, inverse Compton losses drop off to a logarithmic energy
dependence due to Klein-Nishina suppression, while synchrotron losses maintain bsync ∝ E2e .
Thus synchrotron losses dominate over Compton at TeV and higher energies.
The losses themselves evolve with redshift, through the dependences on interstellar den-
70
sities, i.e., the ISRF energy density Uisrf , the interstellar magnetic energy density Umag =
B2/8π, and the number densities nism of interstellar particles. The magnetic field couples
to all cosmic rays, for which the ions dominate the energy density. This coupling is likely
responsible for the approximate equipartition of magnetic field and cosmic-ray energy den-
sities observed in the local interstellar medium. We thus assume that the magnetic field
energy density scales with the cosmic-ray ion flux, and in turn with the star-formation rate:
Umag ∝ S(z). For our fiducial normal galaxy model, we use B0 = 4 µ G, to match the model
of Strong et al. (2010), which is comparable with the typical value in Strong et al. (2011).
We take interstellar particle densities, which control bremsstrahlung losses, to scale as
ni(z) = ni,0(1 + z)3 (3.18)
This can be viewed a consequence of the disk radius scaling with the cosmic scale factor
Rdisk ∝ a ∝ (1 + z)−1 in the Fields et al. (2010) model, and to reflect proportionality
between galaxy disk and dark halo sizes. The fiducial number densities of neutral and ionised
hydrogen, i.e. nH II,0 = nH I,0 = 0.06 cm−3. The helium content of ionised and neutral gas
is also included, with yHe = 0.08. These parameters are broadly consistent with values at
intermediate heights h above the Galactic plane, as befits the volume average represented
by our one-zone model. For reference, the values of nism and B used at the Galactic centre
are much higher at B0 = 8.3 µG and nH II,0 = nH I,0 = 0.12 cm−3.
For starburst galaxies, the magnetic fields and the interstellar densities are higher. The
uncertainty attached with them are also higher compared to Milky Way type normal galaxies.
The fiducial values for magnetic field chosen here is B = 300 µG, although different values
are possible depending on the radio observations of electrons and assumptions regarding
the e/p injection ratio (Lacki et al., 2010; Paglione & Abrahams, 2012; Lacki et al., 2012).
Changing B will change the importance of the IC relative to synchrotron and also pionic
losses. The fiducial ISM density is nism = 600 cm−3 is comparable to the mean densities
71
from observations of M82 (Weiß et al., 2001). These parameter values are in similar range as
used by other groups such as Lacki et al. (2012); Paglione & Abrahams (2012). The redshift
dependence of these parameters is chosen to be the same as for normal galaxies.
3.5 Inverse Compton Emission from Individual
Star-Forming Galaxies
As seen in eq. (3.2), we first compute the luminosity of single galaxy and then average suitably
over the luminosity function to get the luminosity density or cosmic volume emissivity. In
this section, we address the former, focussing first on the IC emissivity within the volume of
a single galaxy, then using this to compute the total luminosity from that galaxy.
3.5.1 Inverse Compton Emissivity of a Star-Forming Galaxy
The specific IC volume emissivity within a galaxy is the rate of producing gamma ray
photons, dΓe−γ→e−γ/dEem per background photon, multiplied by the ISRF number density,
nisrf
dqγdEem
=dNγ
dV dEemdt=
∫
dǫdΓe−γ→e−γ
dEem
dnisrf
dǫ. (3.19)
Here, the gamma ray rate per unit ISRF photon is a product of the cosmic ray flux and the
IC cross section
dΓe−γ→e−γ
dEem=dσe−γ→e−γ
dEemφe . (3.20)
that in general takes the Klein-Nishina form. The full Klein-Nishina cross section is a com-
plicated function of the energies involved. Moreover, the inverse Compton process produces
anisotropic emission if either the cosmic-ray or ISRF populations depart from isotropy. While
72
the cosmic-ray electrons of interest are well-approximated as isotropic, in a real galaxy the
photon field is certainly anisotropic as well as spatially-varying. These effects are included
in GALPROP (Moskalenko & Strong, 2000), but we will ignore them in our simple one-zone
galaxy model. In general, an anisotropic ISRF leads to an overall anisotropy in IC 3. From
figures 5(a) and 6(a) of Moskalenko & Strong (2000), the ISRF anisotropy is a ∼ 20% effect
in the regions of strong IC emission, i.e., within the diffusion length of a few kpc of the disk.
Thus, this effect will not dominate our final error budget.
For the electron spectra we consider, the Thomson limit is a good approximation for
CMB and IR photons, but fails for UV and some optical photons. Hence, we use the Klein-
Nishina result via the prescription of Jones (1968) and Blumenthal & Gould (1970). This
differential cross section is expressed as,
dσ
dEem(ǫ, Ee, q) =
3
4
σT
ǫ γ2
(
2q ln(q) + (1 + 2q)(1 − q) +1
2
(∆ǫq)2
1 + ∆ǫq(1 − q)
)
where the electron Lorentz factor γ = Ee/me and σT is the total Thomson scattering cross
section and q appears in eq. (3.17). In the Thomson limit, the last term in the above equation
become negligible.
The inverse Compton volume emissivity within a one-zone galaxy is thus
dqγdEem
=
∫
dǫdΓe−γ→e−γ
dEem
dnisrf
dǫ=
∫
dǫdnisrf
dǫ
∫
dEeφe(Ee)dσe−γ→e−γ
dEem(3.21)
We evaluate the emissivity numerically, by using the ISRF, cosmic ray flux density, and the
cross section mentioned above, §3.3 and §3.4. That ISRF density nisrf is the same as in the
inverse Compton energy loss equation eq. (3.19), so that we calculate the two self-consistently.
3In principle, therefore, a galaxy’s IC emission is anisotropic and depends on the galaxy orientation! Ofcourse, the EGB signal will average over many galaxies and thus over all orientations.
73
It is suggestive to recast the emissivity in eq. (3.21) in the form
dqγdEem
= Uisrf
∫
dǫdnisrf
dǫǫ(
1ǫ
dΓ(ǫ,Eem)dEem
)
∫
dǫdnisrf
dǫǫ
≡ Uisrf
⟨
1
ǫ
dΓ(ǫ, Eem)
dEem
⟩
= Uisrf
⟨
1
ǫ
qe(> Ee)
b
dσ
dEem
⟩
(3.22)
We see that the emissivity can be understood as an effective average over the ISRF en-
ergy density distribution. Here, the differential and total ISRF energy densities are uisrf =
ǫ dnisrf/dǫ and Uisrf =∫
dǫ uisrf respectively. We see that the amplitude of the gamma-ray
emissivity, qγ scales as
qγ ∼ Uisrf
bqe ∼
bicbqe
and in the Compton dominated regime simply as qe and is independent of the ISRF energy
density, Uisrf . This is a statement of calorimetry. Of course, the gamma-ray spectral shape
does remain sensitive to the ISRF even in the case of perfect calorimetry.
In reality, synchrotron and bremsstrahlung compete with inverse Compton losses at dif-
ferent energies, as seen in §3.4. In the Thomson regime, the ratio of synchrotron to inverse-
Compton losses
bsync
bic
Thomps−→ UB
Uisrf
= 0.4
(
1.1 eV cm−3
Uisrf
)(
B
4µG
)2
, (3.23)
is energy-independent; here the numerical factors are for normal galaxies. For our fidu-
cial magnetic field value, this ratio is not far from unity and hence, the shape does not
change much for different parameters describing the ISRF and the magnetic field. On
the other hand, for Ee >∼ 1 TeV, the IC losses are in the Klein-Nishina limit and grow
only logarithmically in energy; then the synchrotron losses dominate In this case, we have
qγ ∼ (Uisrf/Umag)qe. Thus, for high-energy cosmic-ray electrons, the calorimetric approxima-
tion increasingly breaks down.
For starbursts, the syncrotron-to-IC ratio (eq. 3.23) is bsync/bic ≃ 6 in our fiducial model.
Thus, synchrotron losses dominate IC for all electron energies for such high magnetic fields.
This decrease in emissivity due to synchrotron losses is partially compensated by the in-
74
creased star formation rate in starbursts.
The competition among losses depends on the way the interstellar densities evolve with
redshift. Short-lived, massive stars dominate most of the ISRF except possibly for some of
the optical range, and so the energy density scales with the star-formation rate Uisrf ∝ S(z).
Due to cosmic-ray ion equipartition, we take Umag ∝ S(z) as well, and thus the Uisrf/Umag
ratio doesn’t change dramatically throughout the cosmic history. Similarly, we also include a
strong evolution nism ∝ (1+ z)3 evolution to the interstellar particle density, which increases
rapidly towards z ∼ 1, roughly in step with the cosmic star-formation rate. Thus, the
inverse-Compton/bremsstrahlung ratio of energy losses remains roughly constant as well out
to z ∼ 1. The net result is that the spectral shape of inverse Compton emission does not
evolve dramatically with redshift in our model.
3.5.2 Total Inverse-Compton Luminosity from a Single Galaxy
As seen in §3.2, knowing the specific inverse Compton emissivity dqγ/dEem within a single
star-forming galaxy, we can calculate the specific IC luminosity of the galaxy. Namely,
Lγ(Eem) =dNγ,ic
dt dEem=
∫
dqγdEem
dVISM (3.24)
and we have seen that qγ ∝ (bic/btot)qe. Following Fields et al. (2010), we take supernovae
as the engines of cosmic-ray acceleration. Thus we scale the electron injection rate with
the supernova rate, i.e., we have Lγ ∼∫
qe dVISM ∝ RSN ∝ ψ. This linear proportionality
captures the main dependence of IC emission on star-formation rate.
The single-galaxy IC luminosity takes the separable form
Lγ(ψ, z) =ψ
ψgalLgal(z) (3.25)
where we have suppressed the energy dependence for clarity. Here we explicity display the
dominant sensitivity to star-formation, via the overall linear proportionality to the star-
75
Low B
Low electron cutoff
High injection index
Figure 3.2: The inverse-Compton gamma-ray luminosity spectrum for our single-galaxytemplates. (a) Left: Milky-Way template for normal galaxies. The solid black curve repre-sents our fiducial model, with an ISM density nH I,0 = nH II,0 = 0.06 cm−3, magnetic fieldB0 = 4 µG with a cutoff in the cosmic ray spectrum at 2 TeV. This is for the ISRF consistentwith (R, h) = (0, 5) kpc. This model is designed to provide a good fit to the GALPROP PlainDiffusion model of Strong et al. (2010, dot-dashed blue curve). The solid red curve showsthe effect of reducing B to 2 µ G. The dotted red curve is for nH I,0 = nH II,0 = 0.03 cm−3.The red dashed is includes the effect of non-thermal lines in the ISRF. The solid blue showsthe single galaxy template for the Galactic Centre, with nH I,0 = nH II,0 = 0.12 cm−3, andB0 = 8.3 µG. Comparison of the Diffusive reacceleration case with GALPROP with ourone-zone template that fits it is shown in green. The black dotted curve shows effect ofthe modifying the optical component of the ISRF to include effect of finite lifetime of stars.(b) Right: M82 template for starburst galaxies. The fiducial curve for M82 type starburstis shown in black with nH I,0 = nH II,0 = 300 cm−3, B = 300 µG, and a cutoff at 2 TeV.And the ISRF is shown by the thick black curve in Fig. 3.1. And the magenta curve herecorresponds to the ISRF with higher IR temperature also shown in magenta. Variations inB, n, and spectral index are shown by the blue, green and red curves respectively.
76
formation rate. The Milky-Way model LMW(z) is for a galaxy with the present-day Milky-
Way star-formation rate, but allowing for redshift evolution of the interstellar matter and
radiation densities. Starbursts follow similar scalings, but are tied to M82.
Figure 3.2(a) shows our IC luminosity models for z = 0 for normal galaxies, along
with that of Strong et al. (2010). Our fiducial model described in §3.3, §3.4 and §3.5.1
is represented by the solid black curve. The main feature of the inverse Compton E2emLMW
spectrum is that it has a broad maximum corresponding to the break in the electron injection
spectrum at Ee,break = 4 GeV. In our the fiducial model, the optical component dominates
the ISRF, and thus we expect the injection peak to correspond to an IC peak at Eem ≈
γ2ǫ ≈ 100 MeV. This is close to what we find in our fiducial model, and is consistent with
the GALPROP results.
Our fiducial Milky Way model does a good job of reproducing the IC results of Strong
et al. (2010), over most of the Fermi energy range. Thus, our results are equivalent to
normalizing to the GALPROP luminosity for normal galaxies with the Milky Way’s star
formation rate. A fitting function for our fiducial IC model appears below in Table 3.3; the
fit is good to better than 2% over the Fermi energy range. From this fit we see that at 1
GeV the logarithmic slope is around −0.1, so that LMW ∼ E−2.1em . This is considerably flatter
than both the pionic spectrum and the observed EGB at these energies.
Fig. 3.2(a) presents other curves that illustrate the sensitivity of our model to the input
parameters. The resulting trends can be understood in terms of the competition among
the different electron energy losses in the electron propagation through to the resulting IC
spectrum. For example, decreasing the gas density suppresses the bremsstrahlung losses
relative to the total. This increases the ratio of bic/bbrem, which governs the IC spectral
shape at lower energies. The result is a boost in the IC output at lower energies, which also
shifts the IC peak towards lower energies. At high gamma-ray energies beyond ∼ few GeV,
synchrotron losses dominate, and thus the ratio bic/bsync controls the spectrum; thus we find
that reducing the magnetic field increases the luminosity.
77
The choice of ISRF is also important. Our fiducial model adopts an ISRF consistent
with the GALPROP (R, h) = (0, 5) kpc emission, as it best reproduces the IC luminosity
of (Strong et al., 2010). If we change the ISRF to that of the Galactic centre, the IC signal
is amplified, and the peak shifts to lower energies, as shown by the solid blue curve in
Fig. 3.2(a). At the Galactic centre, the magnetic energy density is increased by a factor of
4 relative to the fiducial case, but the photon energy density goes up by more than a factor
of 10. So both the fractions, bic/b and bsync/b are higher, but so is bic/bsync. And so the peak
occurs at a lower energy, but has a greater amplitude. We included the effect of non-thermal
lines in the ISRF, using the GALPROP spectral form. We find that there is only a minor
effect on the galactic luminosity as shown by the red dashed curve in Fig. 3.2(a).
Finally, we show the effect of different redshift evolution of the optical ISRF component.
Namely, the blue dotted curve in Fig. 3.2(a) shows the results if the optical component is
dominated by long-lived stars. In this model the luminosity scales with stellar mass rather
than star-formation rate: Lγ ∝ Lbg ∝M⋆. Compared to our fiducial case, the ISRF shape is
still similar (same red stars), but the redshift history of Lγ(z) ∝ M⋆(z) is different because
ρ⋆(z) builds up more slowly than ρ⋆(z). We see from Fig. 3.2(a) that fixing our single galaxy
template to z = 1 at the peak of star formation activity, reduces the galactic luminosity due
to lower stellar mass.
Thus, there are several uncertainties in our simple model of the gamma ray spectrum for
a single, Milky-Way like galaxy, such as choice of the average ISRF, the electron density, and
the interstellar magnetic field. However, for reasonable interstellar models, these all lead to
modest changes in the normalisation of the luminosity in the Fermi energy range, at most
tens of percent.
Larger changes would be possible if one were to depart from the single galaxy template
in Strong et al. (2010), which we try to reproduce and that is based on a wealth of Milky-
Way data. Allowing oneself sufficient freedom, the inverse Compton EGB signal can be
adjusted by changes, e.g., in the Uisrf/UB and Uisrf/nism ratios that control the degree of
78
electron calorimetry. Of course, this would then drive the system away from the rough
energy equipartition observed in the Milky Way. Even then, large increases in these ratios
would only increase the completeness of electron calorimetry and would raise the IC signal
by a factor <∼ 2; on the other hand, large decreases in these ratios would spoil calorimetry
and could substantially lower the IC luminosity. Thus, the cosmological prediction of the IC
contribution of the star-forming galaxies to the EGB should be fairly robust, unless there
are departures from the equipartition implicit in the Milky-Way-based normalization.
We turn now to the template for starbursts, which we based our models of M82. The
starburst GeV gamma-ray emission shown Fig. 3.2(b) is characterised by the lack of a peak,
because the underlying electron spectrum is a single power law lacking a break. The overall
normalisation of the fiducial model shown by the black curve is essentially a result of the
contest between the higher star formation rate and the lower fraction of IC losses as expressed
by,
Lnormal
LSBG≈
(
ψMW
ψM82
)(
Uisrf,MW
Uisrf,M82
)(
Uisrf,M82 + UB,M82
Uisrf,MW + UB,MW
)
(3.26)
, assuming the acceleration e/p ratio is the same for normals and starbursts. Equation 3.26
predicts that an M82-type starburst show have an IC luminosity that is a factor ∼ 3.6 larger
than our Milky-Way template, which is in agreement with the results show in Fig. 3.2(a)
and (b). Moreover, the differences in the Fig. 3.2(b) curves for different M82 parameters can
be understood in the same way as the same trends in B, nism and ISRF discussed above for
the Milky-Way template.
The M82 template emission at GeV energies is a factor of few higher compared to the
Milky Way template. However, the M82 emission is proportional to its star-formation rate,
which is higher than that of the Milky Way by a factor ∼ 8. Thus, emission per unit star
formation (L/ψ)template ratio is lower for M82 and thus for our starburst template. This ratio
provides the proportionality constant in our scaling law L = (L/ψ)templateψ. In this sense
79
therefore, starbursts are less efficient IC emitters per unit star formation. The major sources
of uncertainty for starbursts are the magnetic field strength, and the injection e/p ratio–if it
is indeed different for starbursts as suggested by Lacki et al. (2012).
3.6 Results: Inverse Compton Contribution to the
Extragalactic Background
Having established a one-zone galaxy template, and explored physically plausible variations
in it, we proceed to compute the IC contribution to the EGB measured by Fermi-LAT from
star-forming galaxies. This is given by the well-known expression
IE =c
4π
∫
(1 + z)
∣
∣
∣
∣
dt
dz
∣
∣
∣
∣
Lγ[(1 + z)E, z] dz (3.27)
with |dt/dz| = (1+z)−1 [(1+z)3Ωmatter +ΩΛ]−1/2H−10 . Here we assume a flat ΛCDM universe
with H0 = 71 km s−1 Mpc−1, Ωmatter = 0.3, and ΩΛ = 0.7, following Fields et al. (2010).
The cosmic IC luminosity density (i.e,. cosmic volume emissivity) is given by substituting
our single-galaxy luminosity of eq. (3.25) into the luminosity function of eq. (3.2). Because
we have Lic ∝ ψ, the IC emission of a star-forming galaxy traces its star-formation rate.
Thus, the IC luminosity function dngal/dLγ at each redshift is proportional to the luminosity
function of any tracer of star formation at that redshift. We explicitly distinguish for both
normal and starbursts over cosmic time. Suppressing the energy dependence for clarity, we
have
Lγ(z) =
∫
Lγdngal
dLγ
dLγ
=Lnorm(z)fnorm(z)
ψMW
∫
ψnormdngal
dLγdLγ +
Lsbg(z)(1 − fnorm(z))
ψM82
∫
ψsbgdngal
dLγdLγ
=ρ⋆,norm(z)
ψMW
Lnorm(z) +ρ⋆,sbg(z)
ψM82
Lsbg(z) (3.28)
80
Fiducial Normal Galaxy modelGalactic Center Normal Galaxy modelFiducial Starburst Galaxy modelTotal IC : Normal + Starburst
Figure 3.3: The inverse-Compton EGB intensity from star-forming galaxies as a functionof energy. The solid blue curve represents our fiducial IC model summing contributionsfrom normals and starbursts shown in dashed blue and green respectively using single galaxymodels of each shown in Fig. 3.2 where interstellar densities evolve with redshift. The dottedblue curve is for an evolving single normal galaxy template with an ISRF corresponding tothe Galactic centre model. The red curve shows the pionic signal from normal star-forminggalaxies only as in Fields et al. (2010). The dashed black curve shows the total contributionfrom normal galaxies ignoring starbursts. The dashed green shows IC contribution of M82type starbursts alone. It is a factor of a few less than that from normals. The solid blackcurve shows the sum total of normal pionic and total star forming IC.
81
where the ρ⋆ = 〈ψ ngal〉 =∫
ψ dngal/dLγ dLγ is the cosmic star-formation rate. Here fnorm(z)
is the fraction of norml star-forming galaxies at redshift, and 1 − fnorm(z) the starburst
fraction. We calculate this as in (Fields et al., 2010), where starbursts are taken to be
galaxies with a star-formation surface densities Σ⋆ > 0.4 M⊙ yr−1 kpc−2 (Kennicutt, 1998);
this gives fnorm(0) ≈ 1 at z = 0, declining as redshift increases, with fnorm(1) = 0.68 at z = 1.
Following Fields et al. (2010) we adopt the Horiuchi et al. (2009a) cosmic star-formation rate,
and a Milky Way star-formation rate ψMW = 1.07M⊙/yr (Robitaille & Whitney, 2010). For
M82, we use ψM82 = 8.0M⊙/yr, consistent with Sanders et al. (2003); Lacki et al. (2012).
The overall amplitude of the cosmic IC luminosity density is thus linearly proportional
to the cosmic star-formation rate. Of course, the detailed spectral shape of the luminos-
ity density reflects redshift dependence of the interstellar matter and radiation encoded in
LMW(z).
Figure 3.3 shows the IC contribution to the EGB for some choices of the Milky-Way
spectrum LMW and also the fiducial M82 template, LM82. We focus first on the results for
the fiducial model, which appear as the solid black curve, and incorporate the full redshift-
dependence of interstellar density encoded in LMW(z). A fitting function for the fiducial
model appears in Table 3.3, valid over the Fermi energy range. For comparison, Fig. 3.3
shows the pionic contribution in red. In our fiducial model, the IC contribution is about
10% of the pionic contribution at the ∼ 300 MeV peak in E2I; growing to about 20% at 10
GeV. This is consistent with our expectations from the order-of magnitude calculations of
§3.2.
The shape of the fiducial IC curve is rather smooth, and rather flat in E2I. This is
because the spectrum it is a redshift-smeared version of the single-galaxy IC emission. This
stands in contrast to the pionic curve that displays its characteristic peak and a steep ∼ E−sp
dropoff at large energies, with sp = 2.75 the propagated proton spectral index.
The bulk of the ficucial IC signal comes from redshifts at the peak of the EGB integrand
in eq. (3.27). This occurs at zpeak ∼ 1. A comparison of the corresponding curves in Figs. 3.2
82
and 3.3 shows that the IC peak in the galaxy luminosity rest-frame energy Eem ≃ 100 MeV
is transformed into an EGB IC peak at ≃ 40 MeV. This translation in energy space is
consistent with redshifting by a factor 1 + zpeak.
IC (Fiducial) : Normal Only
Figure 3.4: A summary of the star-forming galaxy contributions to the EGB intensity, shownas a function of energy. The blue points represent the Fermi-LAT data points (Abdo et al.,2010k). The solid red and blue curves show the gamma ray intensity due to pionic andfiducial IC components from star-forming galaxies as in Fig. 3.3. The solid black curverepresents the sum of the two. The shaded yellow band gives our estimate of the uncertaintyin the predicted total signal, which is dominated by systematic errors that are common tothe IC and pionic components.
Our fiducial model sums the emission from normal and starburst galaxies; the contribu-
tions from each of these star-formation modes are shown separately in Fig. 3.4. The dashed
blue curve shows the EGB signal from normal galaxies, which we see dominates the total IC
contribution. The IC component from starbursts is shown in dashed green, and is lower than
the normal galaxy contribution by a factor ∼ 2− 4. This suppression arises for two reasons.
First, we found that the starburst photon output (L/ψ)template per unit star formation is
83
lower than that of normal galaxies. In addition, the fraction of galaxies that are starbursts
is small today, and while this fraction grows with redshift, it is still subdominant at z ∼ 1
where the EGB contribution peaks.
The dashed black curve in figure 3.3 represents the total contribution of normal galaxies
to the EGB, summing their pionic and IC contributions. Adding the starburst IC model
leads to the solid dashed curve showing minor increase at the energies beyond 100 GeV. Now,
this computation does not include contribution of pionic emission from starbursts, which has
large uncertainties, (eg., Lacki et al., 2012; Paglione & Abrahams, 2012). Indeed, some have
argued that pionic emission from starbursts makes a small contribution to the EGB (e.g.,
Stecker & Venters, 2011), some models predict starburst contributions should saturate the
observe EGB (e.g., Lacki et al., 2012). It is worth mention that the uncertainties in the
starburst IC contribution are smaller than in the pionic case, largely because the electron
propagation is controlled by calorimetry in both the normal and starburst cases. Of course,
the starburst IC contribution is still rather uncertain, particularly if the the e/p injection
ratio in starbursts is much higher that observed in the Milky Way (Lacki et al., 2012; Paglione
& Abrahams, 2012).
Figure 3.2 includes curves which illustrate the parameter sensitivy of our IC results. To
give a sense of the effect of the adopted ISRF, we have computed the IC EGB resulting from
an evolving ISRF whose spectral shape is appropriate for the Galactic centre. This model
corresponds to the dotted blue curve in Fig. 3.2(a). Here again, the main result is that the
curves are very similar. In detail, the Galactic centre has a slightly higher ratio Uisrf/Umag of
starlight-to-magnetic energy density. Thus, electron energy losses are more IC dominated,
and finally the overall IC flux is closer to calorimetric and so slightly higher.
Our fiducial model assumes the ISRF components (other than the CMB) are dominated
by short-lived stars and thus have a redshift evolution that scales with the cosmic star-
formation rate. An extreme alternative is that the optical ISRF is dominated by long-lived
main sequence stars whose energy density scales with the stellar mass and thus the integrated
84
cosmic star-formation rate, as in eq. (3.11). Adopting this scenario, we have calculated the
IC contribution to the EGB. For clairty, we have omitted the curve from Fig. 3.3. but it
is somewhat smaller than the fiducial model across all energies. This is easily understood:
at high redshifts the star-formation rate is higher than today, but the stellar mass is lower
and thus offers less optical photons and reduced calorimetry. But the effect is not dramatic;
even in this extreme case, the reduction in the IC signal is less than a factor of two relative
to our fiducial model.
Thus, we see that the IC results do depend on the adopted interstellar densities and
their redshift evolution. But for reasonable choices, the variations in the final IC EGB
result are ≈ 10±0.3. The IC calculation in this sense appears rather robust to the systematic
uncertainties in the modeling.
It is useful to compare the IC luminosity density in eq. (3.28) with that for pionic emission.
As noted above, the cosmic IC emissivity is proportional to the cosmic star-formation rate,
along with a weak sensitivity to the redshift evolution of interstellar densities. In contrast,
the Fields et al. (2010) pionic emission depends on the product of cosmic star-formation rate
and mean interstellar gas mass, via the nonlinear scaling Lπ ∼ ψ1+ω with ω = 0.714. Thus,
the pionic emissivity represents a different moment of the cosmic star-formation rate Lπ =
〈Lπngal〉 ∼ 〈ψ1+ωngal〉. This nonlinear moment gives different results depending on how the
star-formation distribution evolves with redshift. Two extreme limits correspond to: (a) pure
density evolution, wherein galaxies have constant star-formation rates but evolving number
density; versus (b) pure luminosity evolution in which the star-forming galaxy density is
constant but the star-formation rates all evolve with redshift. The pionic curves we show
correspond to the fiducial Fields et al. (2010) pure luminosity evolution case, which gives a
contribution to the EGB larger by a factor ∼ 3 than in the pure density evolution case. In
the IC calculation, the pure luminosity and pure density evolution are degenerate, in that
these models all give the same cosmic star-formation rate and thus the same IC output.
Figure 3.4 represents the central result of this paper, summarizing the EGB predictions
85
for gamma-ray emission from star-forming galaxies in the Fermi energy range. We show the
fiducial IC curve from Fig. (3.3), the fiducial normal-galaxy pionic model of Fields et al.
(2010), and the total EGB intensity that sums these components. Over the 100 MeV to
300 GeV range, the pionic component dominates the normal galaxy contribution. But as
we have seen, the IC curve is a much flatter function of energy. The IC component thus
becomes systematically more important at higher energies. Consequently, high-energy slope
of the total emission is less steep than that of the pionic signal, and closer to the Fermi-LAT
data shown in blue (Abdo et al., 2010k).
Including the IC emission thus improves somewhat the agreement between star-forming
prediction and the observed EGB slope, a weaker point of the Fields et al. (2010) pionic-
only model. Up to about ∼ 10 GeV, the central values of the model fall below the data,
but are consistent within the theoretical and observational error budget (discussed below).
At higher energies, the model underpredicts the data. Our pionic model neglects starburst
galaxies; it may well be that their pionic emission is important at this energy range (e.g.,
Lacki et al., 2011). In addition, other unresolved sources (Strong et al., 1976a,c) such as
blazars (e.g., Padovani et al., 1993; Stecker et al., 1993; Mucke & Pohl, 2000; Stecker &
Venters, 2011) and other active galaxies (e.g., Grindlay, 1978; Kazanas & Protheroe, 1983),
starbursts (Thompson et al., 2007) are also guaranteed to play a role . In any case, star-
forming galaxies clearly are an important contribution to the Fermi EGB signal, and could
well be the dominant component.
Note also that the IC curve does not become important at low energies until far below
the pion bump. Thus, the Fields et al. (2010) conclusions stand: the pion feature should
still remain as a distinguishing characteristic of a significant star-forming contribution to the
EGB. Measurements below ∼ 200 MeV should show a break in the EGB slope if star-forming
galaxies play an important role. Such a spectral feature provides one way to discriminate
between star-forming galaxies and other sources, such as the guaranteed contribution from
unresolved blazars.
86
Table 3.3: Fitting Functions
Form: Y = 10aX3+bX2+cX+d, with X = log10(E/1 GeV)Y a b c d
E2emLnormal(Eem) [GeV s−1] −2.51 × 10−2 −5.03 × 10−2 −9.49 × 10−2 40.246E2Inormal(E) [GeV cm−2 s−1 sr−1] −1.53 × 10−2 −7.38 × 10−2 −0.130 −7.63
E2emLstarburst(Eem) [GeV s−1] −9.16 × 10−3 −7.38 × 10−3 −5.41 × 10−2 40.888E2Istarburst(E) [GeV cm−2 s−1 sr−1] −2.37 × 10−3 −2.59 × 10−2 −6.29 × 10−2 −7.86
Our estimate of the uncertainty in the total normal galaxy emission is represented by the
shaded band in Figure 3.4. The errors in the pionic contribution dominate, and are taken
from Fields et al. (2010). For the IC contribution, the errors are dominated by systematic
uncertainties in the Milky-Way star formation rate ψMW and in the normalization ρ⋆(0) of
the cosmic star-formation rate that are both ∼ 40% (Fields et al., 2010). These factors are
common to the overall normalizations of both the pionic and IC signals, cf. eq. (3.28). The
effect of the uncertainties in the parameters magnetic field and interstellar densities on the
normal IC emissivity is estimated from variation in the curves in Fig. 3.2(a) to be ∼ 50%. We
propagate the errors in IC and pionic emissivities and the common errors in star formation
rates, to construct the yellow band in Fig. 3.4. Because the pionic signal dominates the
total, it also dominates the total error, and the uncertainty in the IC calculation does not
change the result much except at the very highest energies where other systematics may be
important.
We have ignored the effect of intergalactic absorption of the high-energy gamma rays
via γγebl → e+e− photo-pair production in collisions with extragalactic background light
(e.g., Salamon & Stecker, 1998; Abdo et al., 2010e; Stecker et al., 2012). This attenuation
starts to become significant for gamma rays over few tens of GeV. Therefore, beyond these
energies our results will be suppressed by amounts that depend on the optical depth for these
high-energy gamma rays. In this context it is worth noting that Fermi-LAT observations
of the z = 0.9 active galaxy, 4C +55.17 do not show significant absorption of gamma rays
87
up to about 100 GeV (McConville et al., 2011). This object lies within the z ∼ 1 regime
where most of the IC contribution to the EGB occurs; thus we might expect that attenuation
should be mild at Fermi energies. Recall also that at high energies, the electron propagation
should include catatrophic losses from IC interactions, which have not been included in our
calculations.
3.7 Discussion and Conclusions
We have calculated the contribution to the extragalactic gamma-ray background due to
inverse-Compton emission from star-forming galaxies. To do this we model the IC emission
in individual star-forming galaxies, which arises from cosmic-ray electron interactions with
the ISRF. For normal star-forming galaxies, we normalize to the present-day Milky Way IC
luminosity as determined by GALPROP (Strong et al., 2010). For starbursts, we use the M82
template guided by observations (de Cea del Pozo et al., 2009) and other results (Lacki et al.,
2012; Paglione & Abrahams, 2012). We also provide a prescription for redshift evolution of
interstellar matter and energy densities, based on equipartition arguments. Assuming cosmic
rays are accelerated by supernovae implies that a galaxy’s IC luminosity scales as Lic ∝ ψ;
this further implies that the cosmic IC luminosity density or volume emissivity is proportional
to the cosmic star-formation rate: Lγ ∝ ρ⋆.
This linear dependence of IC luminosity with star-formation rate has important im-
plications in light of the star-forming galaxies resolved by Fermi. Namely, the Lic ∝ ψ
trend provides a poor fit for Fermi galaxies, for which the observed correlation is non-linear:
Lic ∝ ψ1.4±0.3 (e.g., Abdo et al., 2010f). This implies that the IC component is subdominant
in Fermi galaxies, adding indirect evidence for the primacy of pionic emission in star-forming
galaxies and prefiguring the trends we find for the EGB.
Turning to the EGB, we find that the IC contribution has a very broad maximum in
E2Iic at E ≃ 40 MeV, falling off very gradually away from this peak. This shape is a
88
redshift-smeared reflection of the underlying Milky-Way-like spectrum from the individual
galaxies. In fact, the IC emission is so broadly peaked that is it effectively featureless. This
contrasts markedly with the distinctive pionic feature from cosmic-ray hadronic interactions
in star-forming galaxies.
The amplitude and shape of the IC contribution to the EGB depends on the nature of
the interstellar radiation and matter fields, and their evolution with redshift. However, we
find that when we adopt different but physically motivated variations to our fiducial model,
the final EGB predictions change relatively little. This rough model-independence of the
IC EGB is a consequence of partial calorimetry, i.e., the fact that IC photons represent
a substantial fraction of cosmic-ray electron energy loss, so that the IC output is tied to
cosmic-ray production via energy conservation.
We find that in all of our models the IC signal is always smaller than the pionic contribu-
tion. However, while this is true, IC also has a substantially flatter spectrum, so that the IC
becomes increasingly important away from the pionic peak. This implies that, at least for
normal star-forming galaxies, the IC contribution should dominate over pionic at high and
low energies. For normal star-forming galaxies, one-zone models for scale heights h < 2 kpc
IC can exceed pionic in a narrow window about TeV energies at the high end and below
100 MeV at the low end. However, at higher energies the opacity of the universe becomes
important and losses become catastrophic, and requires different techniques to handle cor-
rectly. At energies below the Fermi-LAT range, the IC emission will be supplemented by
processes such as bremsstrahlung, which we have not included; a detailed treatment of the
MeV regime appears in Lacki et al. (2012).
The approximate calorimetric relationship between IC photons and cosmic-ray electrons
has important consequences. The main redshift dependence of the cosmic IC emissivity is a
linear dependence on the cosmic star-formation rate. Thus, our results are independent of
whether the cosmic star-formation rate is a result of pure luminosity evolution, pure density
evolution, or something in between. This is in contrast to the pionic case, which depends
89
nonlinearly on the star-formation luminosity function and so breaks the degeneracy between
the pure luminosity and pure density evolution cases.
Our IC results for normal and starburst galaxies share important similarites. In both
cases fractional calorimetry controls the basic scaling with star-formation rate, and leads to
similarly broad spectral features that are not very different from power laws. While the M82
starburst template has a higher gamma-ray luminosity than the corresponding Milky-Way
normal galaxy template, the photon output (L/ψ)template per unit star formation is lower for
starbursts. For this reason, and becuase of the relative rarity of starburst galaxies, the IC
contribution to the EGB is dominated by that of normal galaxies. However, the starburst
model has fewer observational constraints and thus larger uncertainties. The starburst IC
contribution could become very important if, for example, the e/p injectio n ratio is very
large in starbursts Lacki et al. (2012).
To simplify the discussion, the IC and pionic models presented here neglected the effects of
Type Ia supernovae, implicitly assuming instead that all supernovae in star-forming galaxies
are due to core-collapse. Lien & Fields (2012) considered in detail the effect of Type Ia
supernovae on the pionic signal. They found that a self-consistent treatment includes both
the addition of Type Ia explosions as cosmic-ray accelerators, but also as part of the Milky-
Way normalization of the cosmic-ray/supernova ratio. The effects largely cancel, so that in
the end, the net pionic galactic luminosity and EGB does not change substantially. In the
case of IC emission a similar cancellation will occur. The only new contribution of possible
importance comes from Type Ia explosions arising from long-lived progenitors in quiescent
(i.e., not actively star-forming) galaxies such as ellipticals. Lien & Fields (2012) show that
the pionic emission from these systems could be large if they have a substantial reservoir
of hot, X-ray-emitting gas. But supernova rate in these galaxies represents a subdominant
fraction of cosmic Type Ia activity, which itself is substantially smaller than the cosmic core-
collapse rate. Thus, the IC contribution from these systems will be small. And so inclusion
of Type Ia supernovae in a self-consistent way would change our results very little, less than
90
the uncertainties in the model.
There remains room to improve on our model. Future work would benefit from obser-
vational progress in clearly identifying an IC signal from individual star-forming galaxies,
at energies away from the pionic peak. Theoretical work would benefit from a more de-
tailed model for the ISRF and its evolution, and from the use of additional multiwavelength
constraints on the cosmic ray electrons.
More broadly, a solid identification and quantification of the main components of the EGB
remains a top priority for gamma-ray astrophysics and particle cosmology. Extending the
Fermi EGB energy spectrum to both higher and lower energies will provide important new
constraints. At sufficiently high energies, the cosmic opacity due to photo-pair production
must become apparent if the EGB is dominated by any sources at cosmological distances
(e.g., Salamon & Stecker, 1998; Abdo et al., 2010e; Stecker et al., 2012). And at energies just
below those reported in Abdo et al. (2010k), a break in the EGB spectrum is an unavoidable
prediction if the signal is dominantly unresolved pionic emission from star-forming galaxies
(both normal and starbursts). An independent probe of EGB origin lies in anisotropy studies
(Ando & Komatsu, 2006; Ando & Pavlidou, 2009; Hensley et al., 2010; Ackermann et al.,
2012a; Malyshev & Hogg, 2011). Regardless of the outcome, an assay of the EGB components
will provide important new information (and perhaps some surprises!) about the high-energy
cosmos.
3.8 Acknowledgements
It is a pleasure to acknowledge stimulating conversations with Keith Bechtol, Vasiliki Pavli-
dou, Troy Porter, Tijana Prodanovic, Amy Lien, Andy Strong, Todd Thompson, and Toni
Venters. After our paper was first submitted, we benefitted greatly from discussions with
Brian Lacki regarding his similar work. BDF would also like to thank the participants of
the 2012 Sant Cugat Forum on Astrophysics for many lively and enlightening discussions.
91
This work was partially supported by NASA via the Astrophysics Theory Program through
award NNX10AC86G.
92
Chapter 4
Statistics of gamma-ray sources
This chapter is an outlines a work in progress with co-authors Eric Baxter (first author),
Scott Dodelson and Brian Fields.
There is great debate about the relative contribution of various gamma-ray emitting
sources to the diffuse EGB. The way to resolve this is to study the various distinguishing
properties of these sources as their gamma-ray spectral slope ,angular anisotropy in the
source distribution, correlation with other wavelengths, etc. The statistics of the photons
themselves bear a signature of the sources producing them. There are a number of candidates
that contribute to the diffuse EGB. AGNs, star forming galaxies, pulsars, etc. are examples
of known astronomical sources. In addition, there is possibility of unknown or exotic sources
contributing to the EGB. The most exciting possibility is the contribution from particle
dark matter via decays and annihilations producing gamma-rays. The known astronomical
sources would be foregrounds to this. Thus it is, crucial to model them accurately not only
in order to understand them but also to seperate them from a possible signature of dark
matter.
For e.g., AGNs are fewer and brighter individually. Amongst these are blazars, AGNs
whose jets point towards us. They are the dominant source class in the gamma-ray emitting
point source catalog (Nolan et al., 2012a). Star forming galaxies on the other hand are far
more numerous, but individually dim. Thus, one expects them to have a different photon
distribution. The gamma-ray emissivity, qγ of a distribution of sources can be given as,
qγ = nγLγ (4.1)
93
Blazars have a number density of nγ ∼ 10−9Mpc−3 and an average luminosity, Lγ ∼
103photons sec−1. Compared to this the star forming galaxies on the other hand have
nγ ∼ 10−3Mpc−3 and Lγ ∼ 106photons sec−1. This rough estimate suggests that the emis-
sivity from a distribution of few, bright blazars is comparable to that from a distribution of
numerous, but dim star forming galaxies.
A detailed analysis of statistics of source photon counts distribution however, has the
potential to reveal the true relative contribution of these sources. This is undertaken in
Baxter, Chakraborty, Dodelson and Fields,.(in prep). An method based on extension of the
P(D) analysis described in (Scheuer, 1957) is used to analyse the photon counts from Fermi
-LAT. The quantity of interest here is the probability Pi(C) of finding Ci photons in the
pixel i of the sky. The individual Pi(C) of the different contributing source types here are
determined from their redshift-dependent luminosity functions. The luminosity functions of
five different source types including the two classes of gamma-ray emitting blazars, are used
namely
1. Star forming galaxies (SFGs)
2. Flat Spectrum Radio Quasars (FSRQs)
3. BL Lacertae Objects (BL LACs)
4. Dark matter annihilation from Galactic subhalos
5. Diffuse emission from the Galaxy
4.1 Formalism
From the luminosity function dndL
of a source type, the differential source flux distribution
dndF
is computed. This is an observed quantity. Of course, what is really computed is the
94
probability of observing flux F, from a single source along the line of sight, P1(F ) as,
P1(F ) ∝∫
dzdLd2
L
(1 + z)2E(z)
dnc
dLδ
(
F − L(1 + z)−(α−2)
4πd2L
)
(4.2)
=
∫ z(Lmax,F )
z(Lmin,F )
dzd4
L(1 + z)(α−2)
(1 + z)2E(z)
dnc
dL
∣
∣
∣
∣
L=4πd2L
F (1+z)(α−2)
(4.3)
where dL is the standard luminosity distance. And the delta function ensures that the flux is
F for a source of luminosity L at dL. The underlying assumption here is that sources have a
power law spectral energy distribution, an assumption good enough for star forming galaxies
and blazars in the regime of interest. z(Lmin, F ) and z(Lmax, F ) are the redshifts expressed
as functions of the minimum and maximum, luminosity Lmin and Lmax, respectively, that
produce observed fluxes F . The source flux distribution dndF
is related to this by,
dn
dF= µP1(F ) (4.4)
where µ is the mean number of sources per unit solid angle along the line of sight.
Having determined P1,a for each source type a, the probability of measuring a sum total
flux F of all the source types, i.e. the flux probability distribution is the convolution of the
individual P1,a.
P (F ) = P1,a ⋆ P1,b..... ⋆ P1,z (4.5)
In reality LAT observations constitute discrete photon counts. The continuous flux probabil-
ity is related to the discrete photon count probability in a particular pixel i via the exposure
in that pixel as
Pi(C) =
∫
dFPi(F )exp(−EiF )(EiF )C
C!(4.6)
and analogous to the continuous flux distribution of a number of sources as in eq. (4.5), the
95
discrete counts distribution is also a convolution of the individual source types,
Pi(C) =C∑
J,K,L,M=0
P FSRQi (J)P SFG
i (K)PDMi (L)PBLLac
i (M)
PGali (C − J −K − L−M) (4.7)
Thus, the data i.e. the counts distribution puts constraints on the model parameters in the
luminosity function of the source types.
96
Chapter 5
High Energy Polarisation
The work in this chapter was done in collaboration with co-authors Vasiliki Pavlidou and
Brian Fields. It is yet to be submitted to a journal.
5.1 Introduction
Active galaxies or AGNs (active galactic nuclei) are some of the most luminous yet mysterious
astronomical objects in the universe. Their particle and radiative emissions are powered by
central supermassive black hole accreting matter. Part of the gravitational energy associated
with accretion is converted into energy of particles such as cosmic rays and neutrinos and
high energy radiation like X-rays and gamma-rays. Thus, these particles and radiation are
messengers of the extreme astrophysical conditions in the core of active galaxies.
Blazars are distant AGNs where the observer’s line of sight is along the jet axis, i.e.
the observer looks down into the jets. Polarised, high energy photons are expected to be
produced by inverse-Compton scattering of low energy seed photons in blazars (McNamara
et al., 2009; Krawczynski, 2012). Seed photons can come from external sources (Meyer et al.,
2012) like the accretion disk and cosmic microwave background (CMB) and internal sources
within the jet like the synchrotron photons (Zacharias & Schlickeiser, 2012) emitted by the
electrons. External target photons through external Compton (EC) mechanism are expected
to produce a lower degree of polarisation than internal photons such as self-synchrotron
Compton (SSC) (McNamara et al., 2009; Krawczynski, 2012).
Various properties of the radiation from blazars like the overall intensity, spectrum, and
97
variability have been studied with multiwavelength observations. However, polarisation at
high energies has received much less attention, not just for blazars, but for any source
class. High energy polarisation measurements of the Crab nebula / pulsar (e.g., Dean et al.,
2008) are the most significant of the high energy polarimetric observations with the Crab
being used for calibrating polarisation observations. There’s also evidence for solar flares
producing polarised X-rays (McConnell et al., 2003). Amongst transient sources, there are
very few successful observations of gamma-ray bursts (GRBs) as listed in Chang et al.
(2013) that . No high energy polarimetric data exists for blazars. This is in part due
to the challenges in measurement of polarisation in X-rays and soft gamma-rays. However,
with numerous X-ray and soft gamma-ray polarimeters at various stages of planning, design
and operation and studies of optical / FIR polarisation properties of blazars underway, a
systematic study of high energy polarisation from blazars is due. Thus, the amount and
degree of high energy polarisation from blazars is an empirical and open question. However,
the theoretical expectations are described in section 5.3. Throughout when referring to high
energy polarisation, we mean X-ray and soft gamma ray polarisation.
Of course, polarised emission from GRBs and blazars at low energies has been studied in
a lot more detail (e.g., Lazzati, 2006; Rossi et al., 2004; Wardle & Kronberg, 1974; Agudo
et al., 2010; Fujiwara et al., 2012). Synchrotron emission is intrinsically polarised (Rybicki
& Lightman, 1986) extending over a broad range of energies and is the main emission mech-
anism in GRB prompt and afterglow emission as well jet emission from both GRBs and
active galaxies. The polarisation depends on magnetic field structure in the environment as
well as the energy distribution in the jets (Lazzati, 2006). Blazars have two broad peaks in
the spectral energy distribution of the jet emission. The first, lower energy peak is due to
synchrotron emission that is intrinsically polarised (Rybicki & Lightman, 1986). It ranges
from radio to as high an energy as X-rays. Observed emission from blazars at lower energies
show polarisation (e.g. Fujiwara et al., 2012). The second peak is typically associated to
inverse-Compton scattering of low energy photons that retains a fraction of this polarisa-
98
tion due to the nature of the inverse Compton scattering process (Bonometto et al., 1970;
Krawczynski, 2012).
Polarisation is a key ingredient in the multimessenger understanding of blazars. The
gains of studying this property are enormous. Polarisation can distinguish between different
emission mechanisms of blazars such as different leptonic models and potentially between
hadronic and leptonic. The connection with the hadronic models is mainly through the
secondary leptons. Hadronic models are accompanied by neutrino emission, whereas polari-
sation should predominantly be from leptonic models via Compton scattering. So there may
be an interesting neutrino polarisation connection to explore. Existence of polarisation in X-
ray and gamma-ray emission from blazars distinguishes it from unpolarised sources at these
energies. Furthermore, if blazars were to possess a reasonably high degree of polarisation,
being high redshift sources like gamma-ray bursts, they could be used for probing Lorentz
invariance violation using vacuum birefringence i.e. the rotation of the plane of polarised
light due to quantum gravity models (e.g., Coleman & Glashow, 1999; Ellis et al., 2000;
Laurent et al., 2011). The variability of blazars is a plus which can serve as a diagnostic for
monitoring the relation between the jet activity and polarisation. This will also allow mea-
surement of polarisation during the flaring states of those blazars that would under normal
circumstances be undetectable in polarisation.
In this paper, we wish to focus on detection prospects of X-ray and soft gamma-ray
polarisation of blazars with polarised seed photons in their jet. In Sec. 5.2, we discuss the
degree of polarisation expected from inverse-Compton scattering of polarised low energy
photons by relativistic electrons in the jet with a power law distribution. The minimum
detectable polarisation (MDP) for various telescopes in general is reviewed with detectability
of a typical blazar in Sec. 5.3. lists some of the brightest blazars and the time required to
detect the MDP for different polarimeters. The detection prospects are improved due to
flaring as described in section 5.4. In the section 5.5, we discuss the potential of polarised
data to indirectly constrain hadronic emission. And in section 5.6, the conclusions along
99
with future directions are described.
5.2 Degree of polarisation of blazar
The degree of polarisation is the fraction of polarised light. Polarisation may be linear,
circular or more generally elliptical. In general, the extent of polarisation is quantified in
terms of the Stokes’ parameters, I, Q, U, V where as usual
I = Ix + Iy
Q = Ix − Iy
U = Ia − Ib
V = Ir − Il (5.1)
a, b represent directions at 45 to x, y respectively. I is the intensity of light, Q and U are
measures of linear polarisation along any direction say x and along 45 from x respectively
(along a) and V is the measure of circular polarisation. Q = 100% implies that the light is
polarised along the + or - x-axis. If Q = U = 0, then V = I and light is circularly polarised.
The degree of polarisation is in general given by,
Π =
√
Q2 + U2 + V 2
I(5.2)
The degree of linear polarisation consists only of Q and U, Πlin =
√Q2+U2
I. Bonometto et al.
(1970) calculated the polarisation of photons produced by inverse-Compton scattering of
an arbitrary distribution of photons off an arbitrary, distribution of electrons. The simpler
case of monochromatic beam of photons is studied that sets up future papers to study more
realistic and complicated distribution of photons. The electrons are unpolarised. Both the
target photons and the electrons are assumed to not have any spatio-temporal variations.
There are two key conditions central to this calculation relating the initial ǫ and scattered
100
photon energies Eγ. The first is the definition of inverse-Compton scattering,
Eγ ≫ ǫ (5.3)
The second is the Thomson limit in the center of mass frame, valid for a lot number of
astrophysical scenarios including that for blazars,
Eγ
me
ǫ
me
≪ 1 (5.4)
For the inverse-Compton upscattered radiation, Bonometto et al. (1970) find the degree of
linear polarisation for completely (100%) polarised target photons is computed to be
ΠBCS =Σ1 + Σ2
Σ1 + 3Σ2(5.5)
where the Σ’s are related to the electron distribution function, m(γ) as a function of the
Lorentz factor, γ,
m(γ) =dne
dγγ−2 ; Σ1 =
∫ 1
0
m(γ)
(
x2 − 1
x2+ 2
)
dx ; Σ2 =
∫ 1
0
m(γ)(1 − x)2
x2dx (5.6)
ne is the number density of electrons and x = γmin/γ, with γmin being the minimum Lorentz
factor required by the electron to produce an up-scattered photon of a given energy given
by γmin = Eγ
2 me(1 +
√
1 + 4 m2e
ǫEγ). Furthermore, their results are computed for an isotropic
distribution of electrons. This is a good assumption as the relativistic electrons under consid-
eration here scatter photons into a narrow cone ∼ γ−1 around the direction of propagation.
And as long as the electron distribution doesn’t vary a lot over this solid angle in the rest
frame of the electron plasma, the results hold true.
According to Bonometto et al. (1970), unpolarised target photons lead to unpolarised
light post inverse-Compton scattering. This can be understood in terms of their observation
101
that Stokes’ parameters change signs in changing from one polarisation state to its orthogonal
state as evident from eqns 7.3 − 7.5 in their paper. As a result, unpolarised target photons
that can be viewed as a superposition of equal proportions of orthogonal polarisation, cancel
out to give zero net polarisation. This is in contrast to Thomson scattering, that can produce
polarisation from unpolarised photons.
Krawczynski (2012) re-evaluates analytically and numerically the calculation of polari-
sation due to inverse-Compton scattering. The numerical calculation is required to unam-
biguously ascertain the range of validity of results and also for precision in determining the
degree of polarisation. The degree of polarisation is explicitly computed for the case of
photons scattered in the jet of blazars verifying the result that unpolarised target photons
lead to unpolarised scattered photons. The self-synchrotron and external Compton cases are
studied explicitly with the conclusion that the external Compton leads to a smaller degree
of polarisation owing to low degree of polarisation of target photons either from the start as
for the CMB or by averaging over axisymmetric configuration in case of broad line region
clouds. The degree of linear polarisation from scattering of the mono-energetic unidirectional
photons off a power-law distribution with index p of electrons is given by,
Πpl =(1 + p)(3 + p)
(11 + 4p+ p2)(5.7)
For p = 2, Πpl ≈ 65% and p = 3, Πpl ≈ 75%. Now, Compton scattering of polarised
light leads to polarised light, and the degree of polarisation Π, is modified from the initial
polarisation, Πin, by this above mentioned factor as (Bonometto et al., 1970; Krawczynski,
2012) for a non-thermal electron distribution,
Π = Πpl × Πin (5.8)
Theoretically, for a power-law distribution of electrons, the degree of polarisation of syn-
102
chrotron emission is given by (Rybicki & Lightman, 1986),
Πsync =p+ 1
p+ 7/3(5.9)
that would give a large values for typical p = 2 − 3. However, various effects like Faraday
rotation and inhomogeneities in the magnetic field can reduce this value 1. In light of
these complications, it is more reliable to use the observed values of Πin. We thus, plan
to test empirically the relation between degrees of low energy polarisation and high energy
polarisation. From Fujiwara et al. (2012), more than half of the gamma-ray blazars are
highly (> 10%) polarised for at least part of the duty cycle. This quick estimate suggests,
that from Krawczynski (2012) and Fujiwara et al. (2012), a degree of polarisation of a few %
at least, for Fermi -LAT blazars in X-rays and soft gamma-rays. According to simulations
in McNamara et al. (2009), the numbers are even higher. However, we will use the results
from Krawczynski (2012) based on the analytical calculations in Bonometto et al. (1970).
These numbers are also more conservative.
Now in general, Π is a function of the direction of target photons and energy of the
scattered photons. Figures 8 and 9 in Krawczynski (2012) show Π to be an increasing
function of the scattered photon energy implying by extrapolation that soft gamma-rays
are likely to be polarised. However, a detailed calculation appropriate at hard X-ray / soft
gamma-ray energies is needed to show this explicitly. Also, the calculation of Krawczynski
(2012) is for uniform distributions of magnetic field, seed photons and electrons. However, in
realistic jet models, a full calculation including their spatio-temporal variation as suggested
by eq.(55) in Krawczynski (2012) should be performed.
1http : //www.astro.uni − wuerzburg.de/ mkadler/scripts/jets ss12/jets hand04.pdf
103
5.3 Minimum Detectable Polarisation (MDP) -
Detection sensitivity of telescopes
In terms of classical electrodynamics, polarisation is defined in terms of the direction of the
electric field vector in an electromagnetic wave. A polarisation analyser is a component of
a polarimeter, that passes light with polarisation vector aligned to a particular direction
or axis, and blocks the light with an orthogonal polarisation detector. Of course, for high
energy radiation as for soft gamma-rays and X-rays, the quantum mechanical picture of
individual photons is more applicable. In the quantum picture, it is the helicity of photons
i.e. the projection of the photon’s spin along the direction of propagation which defines
polarisation. With spin 1, photons have three polarisation states, +1, 0, -1. Of these, only
the transverse components corresponding to the ±1 states are non-zero for a real, massless
photon. These correspond to the right and left handed helicities or equivalently the right
or left circular polarised light. Of course, independent of the basis in which polarisation
is expressed, it is measured by detectors in terms of differences of intensities quantified by
Stokes’ parameters given above in eq. (5.1). Measurements of intensities at different angles is
the basic technique behind measuring polarisation. There are several processes which record
and retain the angular variation of intensity of polarised light such as Bragg reflection,
Compton scattering, photoelectric effect and pair production (Novick, 1975) 2. Of these,
the Compton polarimeters are appropriate for the energy range of photons discussed here,
wherein, the full Klein-Nishina cross-section depends on the angle between the polarisation
vectors of initial and scattered photons (Heitler, 1954).
Whether in the classical picture or the quantum picture, the angular dependence of the
polarisation signal can only be quantified in a statistical sense due to presence of backgrounds
and noise. According to Novick (1975); Weisskopf et al. (2010), for a source producing N
counts, the probability distribution of fractional polarisation say P due to statistical fluctu-
2http : //www.cesr.fr/spip.php?article863
104
ations, measured by an analyser passing one polarisation state, f(Π) = exp
(
−NΠ2/4
)
.
Now a real detector has backgrounds and thus, the polarisation signal is a modulation
a = δΠ over and above the background. Therefore, the probability of measuring this po-
larisation signal a, due to the source in presence of the background is given approximately
by, f(Π) = exp
(
−N(Π − a)2/4
)
. Of course, the polarisation signal will likely have a
modulation phase, which complicates the probability distribution further (Weisskopf et al.,
2010).
Thus, (Weisskopf et al., 2010), polarisation can be manifested as a modulation of a signal
observed in the azimuth φ about the line of sight in direction of the source. The modulation
occurs at an angular frequency of two times the rotation frequency. The signal-to-noise ratio
for N counts in case of Poisson noise is 1/√N . Therefore, amplitude of modulation over a
signal of RS counts sec−1 in presence of a background RS counts sec−1, that has only 1%
chance of being exceeded, or at 99% confidence is (Novick, 1975; Weisskopf et al., 2010),
a1% =RS +RB
RS
nσ√N
(5.10)
where nσ = 4.29 for 99% confidence level. The detector response to polarised light is quan-
tified in terms of modulation factor, µ which is the amplitude for 100% polarised light in
absence of background. Therefore the detectability of a given source of certain strength by
a given detector is quantified in terms of the minimum degree of polarisation it can detect
called the minimum detectable polarisation. The MDP given a source in general is therefore
given by Novick (1975); Weisskopf et al. (2010); Kalemci et al. (2004),
MDP =nσ
µ RS
√
(
2(RS +RB)
T
)
(5.11)
with T being the exposure time.
Now given a detector with a certain modulation factor µ and background RB counts sec−1,
105
eq. (5.11) can be inverted to determine the amount of time required to observe a particular
source producing RS counts sec−1. As described above, blazars are theoretically expected
to have polarised synchrotron emission. However, the exact theoretical computation is com-
plicated by effects like Faraday rotation and inhomogeneities in the magnetic field. These
computations are important and need to be performed, but will not be the focus of this
work. Instead we chose to rely on observations.
The GEMS White Paper Krawczynski et al. (2013) proposes for GEMS like missions a
strategy similar to what we describe below. They emphasise on using archival data on low
energy polarisation in order to motivate observations of high energy polarisation. Their idea,
of course is restricted to GEMS like missions. The study here extends to other polarimeters.
And in addition to archival or published data, we plan to perform synergistic polarimetric
observations at low and high energies of blazars like the ones mentioned below using RoboPol.
3
Blazars are observed to show polarised emission at low energies like at IR wavelengths
(Fujiwara et al., 2012). These blazars are selected from the Fermi -LAT catalog, that is an
excellent all sky representation of blazars. We use these observed polarisation values and
determine the degree of high energy polarisation of a blazar as described in section 5.2. We
invert the above eq. (5.11) to determine how long it would take to detect the chosen blazar.
This exercise is performed for 3 of the brightest blazars with the highest degree of near-IR
polarisation. Indeed, there are several other candidates, however, 3C 454.3, PKS 0048-09
and PKS 1124-186 are the brightest blazars that have the highest measured IR polarisation
from the list in and recorded for various polarimeters in Fujiwara et al. (2012). In addition
a source like 3C 454.3 has shown significant variability Wehrle et al. (2012); Jorstad et al.
(2010). Its flaring behaviour makes it a prime candidate for polarisation measurements as
we will comment on in the next section 5.4.
The required observing times for the various X-ray and soft gamma-ray polarimeters
3http : //robopol.org/
106
are computed for these chosen blazars. The polarimeters are PoGOLITE (25 − 100 keV),
GEMS-like mission (2 − 10 keV), ASTRO-H (50 − 200 keV), POLARIX (2 − 10 keV) and
GAP (50 − 300 keV). The flux of the blazars in units of CGS and mCrab (i.e. in terms of
the flux of Crab in that particular range of energies) is tabulated along with the polarisation
of target photons (Πin) and the required observed times in Tables 5.1, 5.2, 5.3, 5.4 and
5.5.
Source Flux Flux Input Polarisation % Observation time(×10−6ergs cm−2 sec−1) mCrab Πin (Flight Times)
3C 454.3 1.85 × 10−4 13.0 10 237PKS 0048-09 4.32 × 10−5 3.04 11.63 3212PKS 1124-186 4.32 × 10−5 3.04 9.45 4861
Table 5.1: Polarisation observational prospects - PoGOLITE: 200 mCrab,10% in 1 flight(20days)
Source Flux Flux Input Polarisation % Observation time(×10−6ergs cm−2 sec−1) mCrab Πin (ksec)
3C 454.3 5.0 × 10−5 3.51 10 3.24PKS 0048-09 5.0 × 10−6 0.35 11.63 239.80PKS 1124-186 9.10 × 10−6 0.64 9.45 2.74 × 103
Table 5.2: Polarisation observational prospects with XRT / ROSAT fluxes - GEMS (2 - 10keV): 1 mCrab, 2% in 1000 ksec
Source Flux Flux Input Polarisation % Observation time(×10−6ergs cm−2 sec−1) mCrab Πin (ksec)
3C 454.3 1.32 × 10−4 9.28 10 214.90PKS 0048-09 4.32 × 10−5 3.04 11.63 1.49 × 103
PKS 1124-186 4.32 × 10−5 3.04 9.45 2.25 × 103
Table 5.3: Polarisation observational prospects with Swift-BAT fluxes - ASTRO-H (50-200keV): 10 mCrab, 10% in 1000 ksec
Source Flux Flux Input Polarisation % Observation time(×10−6ergs cm−2 sec−1) mCrab Πin (ksec)
3C 454.3 5.0 × 10−5 3.51 10 8.03PKS 0048-09 5.0 × 10−6 0.35 11.63 599.42PKS 1124-186 9.10 × 10−6 0.64 9.45 274.90
Table 5.4: Polarisation observational prospects with XRT / ROSAT fluxes - POLARIX(2.0-10.0 keV): 1 mCrab, 10% in 100 ksec
107
Source Flux Flux Input Polarisation % Observation time(×10−6ergs cm−2 sec−1) mCrab Πin (days)
3C 454.3 1.85 × 10−4 13.0 10 4.73PKS 0048-09 4.32 × 10−5 3.04 11.63 97.21PKS 1124-186 4.32 × 10−5 3.04 9.45 64.24
Table 5.5: Polarisation observational prospects - GAP (50.0-300.0 keV): 1 Crab, 20% in 2days
The PoGOLITE results based on our estimates using the analysis in Krawczynski (2012)
suggest that detection of blazars will be challenging with sensitivity of a balloon experiment.
Flaring blazars observed with dedicated balloon flights may have a chance of detecting the
polarisation from blazars. This will demand, planned observations based on triggers from
the optical, UV, IR telescopes monitoring flaring activity. With the future space based
instruments, the sensitivity is much higher and thus, statistically significant detections are
highly likely with reasonable observing times. Now, the space based instruments are not
dedicated merely to polarisation measurements and therefore, observational strategies are
required. Due to the high sensitivity, quiescent or non-flaring blazars are also expected to be
discovered. This is achievable during all sky scans as well as by targeted, longer observations
of the more dim blazars. And for flaring blazars optical, UV, IR triggers can be used for
making detections. These strategies must be kept in mind while planning for future missions.
In addition to using published and archival data, we plan to perform synergistic, mul-
tiwavelength polarimetry with RoboPol 4 and high energy polarimeters. RoboPol is a fast
optical polarimeter designed specifically to study blazars. It is a pointed instrument with
high cadence. The camera onboard the instrument is designed specifically for monitoring of
blazar jets . The polarimeter is mounted on the University of Cretes Skinakas Observatory
1.3 m telescope. From the above results, idea of using a dedicated, specialised polarimeter at
optical wavelengths in conjunction with a high energy polarimeter will significantly improve
the chances of detecting blazars. This is because monitoring the flaring activity of blazars
as described in detail the next section 5.4 can be very useful to improve detection prospects.
4http : //robopol.org/
108
Furthermore, there is evidence that polarisation certainly at low energies is dynamic (Sorcia
et al., 2013). This suggests that given low energy polarisation is ultimately a source of high
energy polarisation, polarimetric variability could be crucial at X-rays and gamma rays too.
Polarimetric variability means that there are times when the degree of polarisation increases
and therefore detectability improves. Also, variability in flux could be connected to that in
polarisation, a connection that would ultimately lead to clues about the underlying mecha-
nisms. Thus, multiwavelength polarimetry sharpen the emission models which connect the
low energy to the high energy emission. Therefore, in order to faithfully probe the emission
models, it is important to do synchronised multiwavelength observations. This is where ded-
icated, specialised like RoboPol can be critical to our understanding of polarisation and in
general emission processes in blazars.
5.4 Flaring
Blazars are known to have flux variations (Nalewajko, 2013). They fluxes can vary by up
to a factor of 10 or 20 when they flare. As discussed in the previous section, flaring activity
of blazars affect the detectability in the polarisation domain. The direct reason is simply
that those blazars that in their quiescent state fall below the detection limit may be pushed
above threshold when they flare. Secondly, the polarisation signal itself may vary due to the
flaring activity. This could in principle reduce the detectability and needs to be determined
observationally. Here we propose a formalism to quantify the former effect, that is the
increase in the detected fraction of blazars due to an increase in their flux or effectively a
lowering of the detection limit by the same factor.
For this we use the formalism of Feldman and Pavlidou (in prep). Every blazar flares
given a long enough time. Feldman and Pavlidou treat the flaring and quiescent blazars as
a single population with two states. Both flaring and quiescent blazars are represented by a
109
single luminosity function. The two states differ by a flaring factor is given by
R =Ff
Fq(5.12)
where Ff and Fq are fluxes in the flaring and quiescent state respectively. This flaring factor
is assumed to be the same for all blazars and uniform in flux.
They find that the source flux distribution fitting the Fermi -LAT data has contributions
from both quiescent and flaring sources. This implies an extrapolation of the distribution
down to the fluxes at keV and MeV energies at which the polarimeters operate. In practice,
this is easy as blazars have a power-law spectrum. And this will be done at the end when
we provide numbers for the fractional increase in the detected blazars.
For simplicity, we demonstrate the formalism in terms of the LAT fluxes above 100 MeV.
Above F ∼ 2× 10−9 photons cm−2 sec−1, the fit to LAT data is in excellent agreement with
the model that has all the blazars flaring. In accordance with Feldman and Pavlidou, if the
duty cycle i.e. the fraction of time a blazar is in a flaring state is χ, then at any given time
the number of blazars per unit flux in the flaring state is related to that in the quiescent
state,
dN
dFq=
1 − χ
χ
dN
dFf= Cq
dN
dFf(5.13)
where Cq = 1−χχ
represents the factor by which the number of blazars in the quiescent state
exceed the number in the flaring state at any given time in the universe. The mean flux 〈F 〉
of blazars, is a weighted average of the flux of blazars in the quiescent state, Fq and that in
the flaring state, Ff .
〈F 〉 = χFf + (1 − χ)Fq (5.14)
Now the all sky source flux distribution is given in terms of the mean flux observed over
110
a long period of time as,
dN
dF= A
(
F
Fb
)−β1
, F ≥ Fb
= A
(
F
Fb
)−β2
, F < Fb
(5.15)
in accordance with (Abdo et al., 2010l). with the parameter values corresponding to the
best fit values for all blazars both FSRQs and BL LACs put together are AFermi = 16.46 ×
10−14cm2 sec deg−2, Fb = 6.60 × 10−8ph cm−2 sec−1 , β1 = 2.49 and β2 = 1.58. The source
flux distribution is normalised here to the break flux, to give A = AFermiF−β1
b = 1.25 ×
105cm2 sec deg−2 ph according to Feldman and Pavlidou.
As a result of flaring, the break flux is shifted to lower value Fb/R. So, the source flux
distribution is modified to include both flaring and quiescent blazars as they may be treated
as a single population given by,
dN
dF
∣
∣
∣
∣
obs
= A
(
F
Fb
)−β1
+ ARCq
(
RF
Fb
)−β1
, F ≥ Fb
= A
(
F
Fb
)−β2
+ ARCq
(
RF
Fb
)−β1
,Fb
R≤ F ≤ Fb
= A
(
F
Fb
)−β2
+ ARCq
(
RF
Fb
)−β2
, F ≤ Fb
R(5.16)
The observed quantity over an integration time T, is of course, the mean flux 〈F 〉. Now,
this leads to two limits, when the observation time is much longer than the period of the
flare, P and vice versa. In the limit that, the observation time is long enough, T ≫ P , all
“flaring events” including multiple flares from the same source are detected. Therefore, the
number of flaring events is given by NuT/P = NuχT/t. In this case, the mean flux is related
111
to the flaring flux as,
〈F 〉 = χFf + (1 − χ)Ff/R (5.17)
=> Ff =R
(R− 1)χ+ 1〈F 〉 = Reff〈F 〉 (5.18)
Reff =R
(R− 1)χ+ 1,
χT
t≫ 1 (5.19)
For typical values of flaring factor, R = 10 and a duty cycle of χ = 1%, Reff = 9.2. For a
longer duty cycle, say χ = 20%, the mean flux remains closer to the flaring flux and therefore
we get a lower Reff = 3.6. Suppose, Nu is the number of sources undetected in the quiescent
state, but that are detected during their flaring state. Therefore,
Nu =
∫ Fsens
FsensReff
dN
dFdF (5.20)
where Fsens is the sensitivity of the polarimeter. Now in principle, computing Nu tells me
the additional number of blazars detected in polarisation. In practice however, from the
fluxes of our candidate blazars in the previous section and the observation times needed to
detect them, it is only the very bright few blazars that will be detected anyway. And so
the bright end of the source flux distribution dNdF
is of interest. Of course, the exact number
of additional blazars detected by the polarimeters will depend on their flux sensitivity (or
MDP) and of course the duty cycle and periodicity of flaring. A comparison of the mean flux
to the sensitivity gives an idea of the increase in detectability. The mean flux is computed
for limit when all blazars originally below the sensitivity, that exceed the sensitivity limit
due to flaring are detected.
Thus, from equation (5.20), the integration range reduces from ∼ 90% to ∼ 72%. Picking
a sensitivity higher than the break flux Fsens = FbReff as most certainly the blazars detectable
are above the break, could potentially almost double of the number of blazars for a duty
112
cycle of ∼ 20%. However, as stated before the actual numbers would depend on a number
of factors including the assumption that all blazars flare. And indeed since there would
be a handful of blazars, instead of a detailed statistical description, individual blazars that
are known to flare and have a high degree of polarisation at the lower energies should be
selected for pointed observations. We expect that the number of blazars detected due to
flaring activity would nearly double.
5.5 Indirect constraints on hadronic emission
As we discussed above, polarisation is a probe of emission mechanisms of blazars. In fact,
polarised X-rays and gamma-rays are primarily a probe of leptonic emission. Amongst
leptonic emission, inverse-Compton scattering by electrons and synchrotron radiation are the
most likely mechanisms of producing polarisation. The blazar spectral energy distribution
consists of two broad spectral peaks or bumps (Abdo et al., 2010m). The lower energy
peak is usually put down to synchrotron emission that is intrinsically polarised (Rybicki
& Lightman, 1986). The higher energy peak is believed to be due to inverse-Compton
scattering of either the synchrotron radiation itself or some other source of photons. As
described above in detail, inverse-Compton emission in blazars is likely to be polarised since,
the underlying target photon distribution is polarised. And data on polarised radiation is
capable of identifying and distinguishing between synchrotron emission and inverse-Compton
emission as they in principle produce different degrees of polarisation. Thus, thus discovery
of polarised emission at high energies would be a test of models of blazar SEDs amongst
other things.
Now high energy emission could also be hadronic (e.g. Becker, 2008; Reimer, 2012; Doert
et al., 2012; Dermer et al., 2007). Hadronic processes involving protons produce high energy
radiation mainly gamma rays in the following ways. They first interact with either other
113
protons or photons (Becker, 2008).
p γ → ∆+ → p π0
→ n π+ (5.21)
p p→ π+ π− π0 (5.22)
The resulting pions decays produce gamma rays directly or indirectly. Neutral pions decay
to photons directly as
π0 → γ γ (5.23)
The charged pions decay into secondary leptons, that can then produce synchrotron or
inverse-Compton emission.
π+ → µ+ νµ → e+ νe νµ νµ (5.24)
Hadronic emission from blazars could also produce polarisation from the secondary lep-
tons. Once, again the leptons would produce IC scattering and synchrotron radiation. The
spectrum of this polarised radiation would be different from that produced by the primary
leptons. As a result, by spectro-polarimetric observations secondary leptonic gamma rays will
be identified and their relative proportion to other processes determined. Particularly as the
secondary leptonic emission can be related to its primary hadronic source through branching
ratios, it is possible to put indirect constraints on the primary hadronic emission via the lep-
tonic secondaries. And as, these leptons are again correlated by well defined branching ratios
to neutrinos, there are some prospects of a possible relation between neutrino and polarised
high energy emission, though it is likely to be model dependent and potentially complicated.
But these are exciting avenues that need to be explored in order to completely understand
those very complications that are so central to the emission mechanisms in blazars.
114
5.6 Discussion and conclusions
It is very clear that polarimetry is the next frontier in high energy astronomy that holds a
lot of promise. It is extremely challenging to perform polarimetric measurements in general.
Specifically for blazars, it is possible that instruments in space are required to make detec-
tions. However, in an ideal situation, i.e. if there are dedicated balloon flights performing
observations of previously selected sources that are flaring, then it is not inconceivable that a
signal maybe be observed. Also, remembering that high energy polarisation hasn’t yet been
measured and the predictions are based on theoretical assumptions albeit reasonable ones,
if the polarisation values happen to be as high as simulations in McNamara et al. (2009)
predict then balloon experiments will have an excellent chance of detecting something.
The strategy of using the low energy polarisation measurements of blazars from Fujiwara
et al. (2012); Blinov et al. (2013) and other similar work to select blazars looks promising
to effectively focus on the ones which are the brightest and most polarised. All of this is
expressed in terms of the MDP and required observation times. This is similar to the plan
proposed by the GEMS team Krawczynski et al. (2013). The numbers for GEMS-like mission
are consistent with those found by This is similar to the plan proposed by the GEMS team
Krawczynski et al. (2013). Of course, this study is not limited to GEMS like missions or that
energy range. Predictions have been made for a number of different polarimetric missions
such as PoGOLITE, Astro-H, POLARIX, GAP.
As discussed in section 5.4, flaring increases detectability of blazars. Monitoring flar-
ing activity of sources in optical and other wavelengths to determine flare statistics will
help build our database of promising candidates while waiting for upcoming polarimetry
missions. Low energy polarimetry missions like RoboPol can be used to guide high energy
polarimetry observations based on monitoring of flaring blazar as Blinov et al. (2013). And in
future, synergistic multiwavelength polarimetry observations will definitely increase chances
of discovering blazars in this new domain.
Polarimetry is critical to not just multiwavelength, but possibly multimessenger astro-
115
physics as pointed out in section 5.5. Neutrinos are indicators of primarily hadronic emission
mechanisms in blazars. Polarisation could be an indicator of both, though it is more directly
related to the primary leptonic emission. It is associated to hadronic mechanism via radia-
tive emission from the secondary electrons. Thus, polarimetry could probe both secondary
and primary emission and hadronic and leptonic emission. And more specifically, it can
distinguish between self-synchrotron and external Compton mechanism.
In addition, high polarimetry of blazars can potentially put constraints on Lorentz Invari-
ance Violation in much the same way as GRBs do. If blazars were to possess a reasonably
high degree of polarisation, being high redshift sources, they could be used for probing
LIV (e.g., Coleman & Glashow, 1999; Ellis et al., 2000; Laurent et al., 2011) using vacuum
birefringence.
Thus, high energy polarimetry is an exciting field with tremendous discovery potential.
There is a huge scope for theoretical work and observational and mission strategising in
preparation for future generation of polarimeters.
116
Chapter 6
Radio Supernovae in the GreatSurvey Era
Abstract
1 Radio properties of supernova outbursts remain poorly understood despite longstanding
campaigns following events discovered at other wavelengths. After ∼ 30 years of observa-
tions, only ∼ 50 supernovae have been detected at radio wavelengths, none of which are Type
Ia. Even the most radio-loud events are ∼ 104 fainter in the radio than in the optical; to date,
such intrinsically dim objects have only been visible in the very local universe. The detec-
tion and study of radio supernovae (RSNe) will be fundamentally altered and dramatically
improved as the next generation of radio telescopes comes online, including EVLA, ASKAP,
and MeerKAT, and culminating in the Square Kilometer Array (SKA); the latter should be
>∼ 50 times more sensitive than present facilities. SKA can repeatedly scan large (>∼ 1 deg2)
areas of the sky, and thus will discover RSNe and other transient sources in a new, auto-
matic, untargeted, and unbiased way. We estimate SKA will be able to detect core-collapse
RSNe out to redshift z ∼ 5, with an all-redshift rate ∼ 620 events yr−1 deg−2, assuming a
survey sensitivity of 50 nJy and radio lightcurves like those of SN 1993J. Hence SKA should
provide a complete core-collapse RSN sample that is sufficient for statistical studies of radio
properties of core-collapse supernovae. EVLA should find ∼ 160 events yr−1 deg−2 out to
redshift z ∼ 3, and other SKA precursors should have similar detection rates. We also pro-
vided recommendations of the survey strategy to maximize the RSN detections of SKA. This
new radio core-collapse supernovae sample will complement the detections from the optical
1This chapter matches the version which has been submitted to Astrophysical Journal, and posted versionon arxiv.org numbered on arXiv:1206.0770 and is co-authored with Brian Fields.
117
searches, such as the LSST, and together provide crucial information on massive star evolu-
tion, supernova physics, and the circumstellar medium, out to high redshift. Additionally,
SKA may yield the first radio Type Ia detection via follow-up of nearby events discovered
at other wavelengths.
6.1 Introduction
Supernovae are among the most energetic phenomena in the universe, and are central to
cosmology and astrophysics. For example, core-collapse supernovae are explosions arise
from the death of massive stars and hence are closely related to the cosmic star-formation
rate and to massive-star evolution, and are responsible for the energy and baryonic feedback
of the environment (Madau et al., 1998). Type Ia supernovae show a uniform properties in
their lightcurves and play a crucial role as cosmic “standardizable candles” (Phillips, 1993;
Riess et al., 1996).
Our knowledge of the optical properties of supernovae, is increasing rapidly with the
advent of prototype “synoptic”–i.e., repeated scan–sky surveys, such as SDSS-II (Frieman
et al., 2008; Sako et al., 2008) and SNLS (Bazin et al., 2009; Palanque-Delabrouille et al.,
2010). These campaigns are precursors to the coming “Great Survey” era in which synoptic
surveys will be conducted routinely over very large regions of sky, e.g., LSST 2 (LSST Science
Collaboration et al., 2009) and Pan-STARRS 3. The number of detected supernovae will
increase by several orders of magnitude in this era (LSST Science Collaboration et al., 2009;
Lien & Fields, 2009).
In contrast to this wealth of optical information, properties of supernovae in the radio
remain poorly understood, fundamentally due to observational limitations. Radio super-
novae (RSNe) have primarily been discovered by follow-up observations of optical outbursts,
and only very rarely by accident. To date, only ∼ 50 core-collapse outbursts have radio
2http://www.lsst.org/Science/docs/SRD.pdf3http://pan-starrs.ifa.hawaii.edu/project/science/precodr.html
118
detections, and no Type Ia explosion has ever been detected in the radio (Weiler et al., 2004;
Panagia et al., 2006). The core-collapse subtype Ibc has been a focus of recent study in the
radio, because some Type Ibc events are associated with long Gamma-Ray Bursts (GRBs)
(Galama et al., 1998; Kulkarni et al., 1998; Soderberg, 2007; Berger et al., 2003).
Current radio interferometers are scheduled primarily around targeted observations pro-
posed by individual principal investigators. This stands in contrast to future radio interfer-
ometers planned for the coming “Great Survey” era. These include the Square Kilometer
Array (SKA4) and its precursor prototype arrays (for example, ASKAP5 and MeerKAT6).
These telescopes will operate primarily as wide-field survey instruments focusing on several
key science projects (Carilli & Rawlings, 2004). As synoptic telescopes, they will be far bet-
ter suited to study all classes of transient and time-variable radio sources, including RSNe.
Gal-Yam et al. (2006) already pointed out the power of synoptic radio surveys for detecting
radio transients of various types, including supernovae and GRBs, in an unbiased way. Here
we quantify the prospects for RSNe.
In this paper we explore this fundamentally new mode of untargeted RSN discovery and
study. We adopt a forward-looking perspective, and consider the new science enabled by
RSNe observations in an era in which the full SKA is operational. Our focus is mainly on
core-collapse supernovae, the type for which some radio detections already exist. However,
we will also discuss the possibility of Type Ia radio discovery based on current detection
limits. We will first summarize current knowledge of radio core-collapse supernovae (§6.2),
and the expected sensitivity of SKA (§6.3). Using this information, we forecast the radio core-
collapse supernovae harvest of SKA (§6.4), and consider optimal survey strategies (§6.5). We
conclude by anticipating the RSN science payoff in this new era (§6.6). We adopt a standard
flat ΛCDM model with Ωm = 0.274, ΩΛ = 0.726, and H0 = 70.5 km s−1 Mpc−1 (Komatsu
et al., 2009) throughout.
4http://www.ska-telescope.org5http://www.atnf.csiro.au/projects/askap6http://www.ska.ac.za/meerkat
119
6.2 Radio Properties of Supernovae
Several key properties of RSNe have been established, as a result of the longstanding lead-
ership of the NRL-STScI group (recently reviewed in Weiler et al., 2009; Stockdale et al.,
2007; Panagia et al., 2006) and of the CfA group and others (summarized in Soderberg,
2007; Berger et al., 2003). We summarize these general RSN characteristics, which we will
use to forecast the RSN discovery potential of synoptic radio surveys.
6.2.1 Radio Core-Collapse Supernovae
Observed core-collapse RSNe have luminosities spanning νLν ∼ 1033 − 1038 erg s−1 at 5
GHz, and thus are >∼ 104 times less luminous in the radio than in the optical. Their intrinsic
faintness has prevented RSN detection in all but the most local universe. Even within a
particular core-collapse subtype, radio luminosities and lightcurves are highly diverse, e.g.,
two optically similar Type Ic events might be radio bright in one case and undetectable in
the other (Munari et al., 1998; Nakano & Aoki, 1997) 7 8. Additionally, core-collapse RSNe
spectral shapes strongly evolve with time; lightcurves peak over days to months depending on
the frequency. RSN emission can be understood in terms of interactions between the blast,
ambient relativistic electrons, and the circumstellar medium (Chevalier, 1982a,b, 1998).
To model RSN emission as a function of frequency and time, we adopt the semi-empirical
form derived by Chevalier (1982a) and extended in Weiler et al. (2002),
L(t, ν) = L1
( ν
5 GHz
)α(
t
1 day
)β
e−τexternal
(
1 − e−τCSMclumps
τCSMclumps
) (
1 − e−τinternal
τinternal
)
. (6.1)
We follow the notation of Weiler et al. (2002). L(t, ν) is the supernova luminosity at fre-
quency ν and time t after the explosion. Optical depths from material both outside (τexternal,
τCSMclumps) and inside (τinternal) the blast-wave front are taken into account (see Weiler et al.,
7http://rsd-www.nrl.navy.mil/7213/weiler/sne-home.html8New Radio Supernova Results are available online at: http://rsdwww.nrl.navy.mil/7213/weiler/sne-
home.html
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2002).
Parameters embedded in each optical depth term are those for SN 1993J, one of the best
studied RSNe (Weiler et al., 2007). Radio emission from SN 1993J is dominated by the
clumped-circumstellar-medium (clump-CSM) term, and hence
L(t, ν) ∼ 1 − e−τCSMclumps
τCSMclumps
, (6.2)
where τCSMclumps= 4.6 × 105 ( ν
5 GHz)−2.1 ( t
1 day)−2.83, for SN 1993J. At small t, τCSMclumps
is
large and L(t, ν) ∼ 1/τCSMclumps∝ ν2.1 t2.83, so luminosity grows as a power law at early
times. With all optical depth parameters fit to SN 1993J, the peak luminosity is controlled
by the prefactor L1.
Our main focus will be on RSN discovery, and thus it is most important to capture the
wide variety of peak radio luminosities, which correspond in our model to a broad distri-
bution for L1. Figure 6.1 shows a crude luminosity function (not-normalized) based on the
sample of 20 core-collapse supernovae (15 Type II and 5 Type Ibc) that have a published
peak luminosity at 5 GHz (Weiler et al., 2004; Stockdale et al., 2003, 2007; Papenkova et al.,
2001; Baklanov et al., 2005; Pooley et al., 2002). We use 5 GHz data to construct our lumi-
nosity function because the most RSNe have been observed at this frequency. However, our
predictions will span a range of frequencies, based on this luminosity function and eq. (6.2).
The data are divided into four luminosity bins of size ∆ log10(L) = 1. The black curve in
Fig. 6.1 is the best-fit Gaussian, with average luminosity log10(Lavg/erg s−1 Hz−1) = 27.3,
a standard deviation σ = 1.25, and χ2 = 0.18. SN 1987A is marked in Fig. 6.1, but was
not used in the fit to avoid possible bias due to its uncommonly low luminosity. The fitted
luminosity function might be biased towards the brighter end, because of the current survey
sensitivity and the small and incomplete nature of the sample.
121
Figure 6.1: Radio luminosity function (not-normalized) at 5 GHz of core-collapse supernovaeshowing core-collapse supernovae count as a function of log10(L), where L is the peak radioluminosity. Data are binned to ∆ log10(L) = 1. The black curve shows the χ2-fitted Gaussianto the underlying data (red stars).
6.2.2 Radio Type Ia Supernovae
All searches to date have failed to detect radio emission from Type Ia supernovae. Panagia
et al. (2006) reported the radio upper limits of 27 Type Ia supernovae from more than two
decades of observations by the Very Large Array (VLA). The weakest limit on a Type Ia
event is 4.2 × 1026 erg s−1 Hz−1 at 1.5 GHz for SN 1987N, which is around one order of
magnitude lower than the average luminosity of radio-detected core-collapse supernovae (see
§6.2.1). The strongest limit on Type Ia radio emission is even tighter, 8.1×1024 erg s−1 Hz−1
at 8.3 GHz for SN 1989B. Additionally, the z ∼ 0 cosmic Type Ia supernova rate is around
1/4.5 of the core-collapse supernova rate (Bazin et al., 2009). The intrinsic faintness in radio
and their smaller rate make detecting Type Ia in radio observations especially hard.
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6.3 Next-Generation Radio Telescopes: Expected
Sensitivity
Radio detections of supernovae to date have been restricted both by the limiting sensitivity
of contemporary radio interferometers and the need for dedicated telescope time for transient
followup. This situation will change drastically with SKA’s unprecedented sensitivity and
particularly by its ability to repeatedly scan large regions of the sky at this great depth.
Current SKA specifications adopt a target sensitivity parameter Aeff/Tsys = 104 m2 K−1
at observing frequencies in the low several GHz, including z = 0 HI at 1.4 GHz. Aeff is the
effective aperture, and Tsys is the system temperature. We will adopt this value of Aeff/Tsys,
which yields a 1-σ rms thermal noise limit in total intensity of
σI = 0.15 µJy (∆ν/GHz)−1/2 (δt/hr)−1/2, (6.3)
for a bandwidth ∆ν and observation duration δt. The SKA will therefore reach a thermal
noise limit of several nJy in deep continuum integrations (δt ∼ 1000 hr) 9. We define the
associated survey sensitivity Smin (the minimum flux density threshold) as Smin = 3σI . In
common with other radio interferometers, SKA will accumulate sensitivity in targeted deep
fields, including transient-monitoring fields, by accumulating integration time over multiple
individual observing tracks. We therefore will adopt a fiducial SKA supernova sensitivity of
Smin = 50 nJy in 100 hours of observation, but we will show how our results are sensitive to
other choices of Smin.
It is anticipated that transient fields will be revisited with a cadence appropriate to
the variability timescales under study and that interferometric inverse imaging methods will
include source models with time variability. Survey optimization for interferometric transient
detection is an active area of current SKA research. The technical details are beyond the
intent and scope of this paper, but will be influenced by science goals for transient source
9http://www.skatelescope.org/PDF/DRM v1.0.pdf
123
study in general, including pulsars, GRBs, and supernovae (as considered in this paper), as
well as the as-yet undiscovered transient population.
6.4 Radio Supernovae for SKA
With its unprecedented sensitivity, SKA will be capable of synoptic search for core-collapse
RSNe and open new possibilities in radio astronomy. In this section, we predict the RSN
detections of SKA based on current knowledge to demonstrate how the RSN survey can be
done.
6.4.1 Core-Collapse Supernovae
The detection rate Γdetect = dNSN/(dtobs dΩ dz) for a given RSN survey is
Γdetect = fsurvey fradio fISM ΓSN , (6.4)
and is set by several observability factors f that modulate the total rate of all supernovae
ΓSN(z) =dNSN
dVcomov dtem
dtemdtobs
dVcomov
dΩ dz= RSN(z) r2
comov(z) c
∣
∣
∣
∣
dt
dz
∣
∣
∣
∣
(6.5)
within the cosmic volume out to redshift z (Madau et al., 1998; Lien & Fields, 2009).
We see that the total cosmic supernova rate ΓSN depends on cosmology via the volume el-
ement and the time dilation terms. Because ΛCDM cosmological parameters are now known
to high precision, these factors have a negligible error compared to the other ingredients in
the calculation. The other factor in ΓSN is the cosmic core-collapse supernova rate density
RSN(z) = dNSN/(dVcomov dtemit). Some direct measurements of this rate now exist out to
z ∼ 1, but the uncertainties remain large (Cappellaro et al., 1999; Dahlen et al., 2004; Cap-
pellaro et al., 2005; Hopkins & Beacom, 2006; Botticella et al., 2008; Dahlen et al., 2008;
Kistler et al., 2011; Bazin et al., 2009; Smartt et al., 2009; Dahlen et al., 2010; Li et al.,
124
2011c; Horiuchi et al., 2011). However, core-collapse events are short-lived, and so the cos-
mic core-collapse rate is proportional to the cosmic star-formation rate ρ⋆, which is much
better-determined and extends to much higher redshifts. We thus derive RSN from the re-
cent Horiuchi et al. (2009b) fit to the cosmic star-formation rate. The proportionality follows
from the choice of initial mass function; we apply the Salpeter initial mass function (Salpeter,
1955) and assume the mass range of core-collapse SNe progenitors to be 8M⊙ − 50M⊙; this
gives RSN = (0.007M−1⊙ ) ρ⋆
Several effects reduce the total rate ΓSN to the observed rate Γdetect in eq. (6.4). Due to
finite survey sensitivity, only a fraction fsurvey of events are bright enough to detect, and only
some fraction fradio of supernovae will emit in the radio. We neglect interstellar extinction
and assume fISM ∼ 1 at the radio wavelengths considered.
The term fradio in eq. (6.4) contains the greatest uncertainty due to the relatively small
sample of RSNe observed to date, and the unavoidable incompleteness of the sample (K.
Weiler, private communication 2010). The only published fraction available is for Type Ibc
supernovae. Using VLA for radio follow-up, Berger et al. (2003) suggests that fradio,Ibc ∼ 12%
after surveying 33 optically-detected Type Ibc supernovae. For the purpose of demonstration,
we will adopt fradio = 10% for the calculations presented in this paper, which we believe is
rather conservative.
An order-of-magnitude calculation provides a useful estimate of the expected core-collapse
RSN rate. As discussed in §6.3, we adopt a fiducial SKA sensitivity of Smin = 50 nJy. Hence
SKA will be able to detect supernovae with average radio luminosity (L ∼ 1027 erg s−1 Hz−1)
to a distance DL =√
L/4πSmin ∼ 4 Gpc, which for a ΛCDM cosmology corresponds to
z ∼ 1. This will give a detectable volume of Vdetect ∼ (4/3)πD3L ∼ 2.85 × 1011 Mpc3.
Observations show that the core-collapse supernova rate RSN ∼ 10−3 yr−1 Mpc−3 at z ∼ 1
(Dahlen et al., 2008, 2010). Assuming the fraction of the total core-collapse supernovae that
display the adopted average radio luminosity to be fradio ∼ 10% (Berger et al., 2003), the
all-sky detection rate dNSN/dt ∼ RSN × fradio × Vdetect ∼ 2.85 × 107 yr−1. This corresponds
125
to a areal detection rate dNSN/(dt dΩ) ∼ 700 yr−1 deg−2. As we now see, a more careful
calculation confirms this estimate.
Figure 6.2: Estimated radio core-collapse supernova detection rate as a function of redshiftat 1.4 GHz, assuming fradio = 10%. Predictions are shown for different survey sensitivities:Smin = 10 µJy (blue), 1 µJy (blue), 100 nJy (blue), 50 nJy (thick red), 10 nJy (blue), 1nJy (blue) from bottom to top solid curves, respectively. We adopt 50 nJy as our bench-mark sensitivity hereafter. For comparison, the red-dashed curve shows the LSST opticalsupernova detection rate per year per deg2 (Lien & Fields, 2009). Also, the top solid curve(black) plots the ideal core-collapse RSN rate for comparison.
A careful prediction involves detailed calculation of fsurvey(z). The fraction fsurvey(z) of
observable radio-emitting events depends on adopted survey sensitivity, and on the normal-
ized supernova luminosity function Φ5GHz(logL), which is measured at a peak luminosity at
5 GHz (derived in § 6.2). In this paper we will only consider whether a supernova is de-
tectable at its peak luminosity at each corresponding frequency. The peak radio luminosity
should be reached earlier at higher frequencies because of preferential absorption at lower
frequencies (Weiler et al., 2002). At different redshift, the peak flux density Speakmin in the
126
Figure 6.3: Estimated radio core-collapse supernova detection rate as a function of redshiftfor different frequency bands, for fradio = 10%, and an adopted survey sensitivity Smin = 50nJy.
Figure 6.4: Core-collapse detection sensitivity to supernova radio luminosity, at 1.4 GHz,and for survey sensitivity Smin = 50 nJy. (a)Left Panel: Supernova distribution over redshift,for different cutoffs for the luminosity function. (b)Right Panel: Supernova distribution inluminosity bins, integrated over all redshifts.
observed frequency ν can be tied to the corresponding luminosity threshold Lpeakmin by
Lpeakmin (z; νemit) =
4πD2L(z)
(1 + z)Speak
min (νobs), (6.6)
127
where the luminosity distance isDL(z) = (1+z) c/H0
∫ z
0dz′ [Ωm(1+z′)3+ΩΛ]−1/2 . However,
because the luminosity function we used is based on the peak luminosity at 5 GHz, we must
find the corresponding luminosity threshold at this frequency by applying corrections based
on the radio spectrum,
Lpeakmin,5GHz = Lpeak
min
∫
5GHz bandSpeak(νem) dνem
∫
obs bandSpeak[(1 + z)νobs] dνobs
. (6.7)
The detectable fraction resulting from survey sensitivity can therefore be estimated as
fsurvey =
∫
log Lpeakmin,5GHz
Φ5GHz(logL) d logL. (6.8)
Figure. 6.2 plots the results of our predicted core-collapse RSN detection rate for different
target survey sensitivities, Smin. We adopt a benchmark frequency of 1.4 GHz because this
will be one of the first major bands SKA deploys to observe neutral hydrogen. The related
instantaneous field-of-view at 1.4 GHz of current SKA designs based on dish reflectors is
approximately 1 deg2, which we adopt. Fig. 6.2 plots the ideal core-collapse supernova
rate for comparison (assuming infinite sensitivity but fradio = 10%). One can see that the
detection rate at 1 nJy is very close to the ideal rate in the universe.
Results for our fiducial SKA flux limit Smin = 50 nJy are highlighted in Fig. 6.2. At this
sensitivity, we see that we can expect that radio supernovae will be discovered (event rates
> 5 RSN yr−1 deg−2) over the enormous redshift range z ≃ 0.5 to 5. The total rate of RSNe
expected in this entire redshift range is
dNSN
dt dΩ(> 50 nJy) ≈ 620 RSNe yr−1 deg−2 , (6.9)
in good agreement with our above order-of-magnitude estimate. This sample size is large
enough to be statistically useful and to allow for examination of the redshift history of RSNe.
Moreover, out to z ∼ 1, SKA will detect almost all cosmic RSNe in the field of view, while
128
at higher redshift the detections still comprise > 10% of the underlying ideal cosmic rate.
For comparison, we also see that LSST will detect optical supernovae out to z <∼ 1. Thus
SKA will be complementary to LSST as a unique tool for cosmic supernova discovery.
Figure. 6.3 shows how core-collapse RSN detections vary for different observing frequen-
cies, fixing a common survey sensitivity Smin = 50 nJy and bandwidth ∆ν = 1 GHz. Results
show similar numbers of detections at different bands, which is caused by a relatively flat
spectrum shape at peak luminosities. Because SKA will be able to detect core-collapse RSNe
out to high redshift z ∼ 5, the frequency-redshift and time-dilation effects are significant.
Weiler et al. (2002) noted that RSNe peak when the optical depth τ ∼ 1. Since the optical
depth depends both on frequency and time with similar power index (Weiler et al., 2002), the
frequency-redshift and time-dilation effects approximately cancel, so that a fixed observed
frequency, the peak time is nearly redshift-independent.
As mentioned above, our luminosity function is likely biased toward the available bright
events in a small and incomplete sample. To explore how this bias could affect our results,
Fig. 6.4 shows how the detection rate with Smin = 50 nJy at 1.4 GHz depends on core-
collapse RSN luminosity. Fig. 6.4(a) shows that RSN with peak luminosities greater than
1027 erg s−1 Hz−1 contribute all of the detections beyond redshift z ∼ 1, and RSN need to
peak brighter than 1028 erg s−1 Hz−1 to be seen beyond z ∼ 3. Fig. 6.4(b) similarly shows that
the all-redshift detection rate becomes substantial for explosions peaking > 1026 erg s−1 Hz−1.
Type Ibc supernovae are of particular interest given their association with long gamma-
ray bursts (GRBs; Galama et al., 1998; Woosley, 1993; Heger et al., 2003). Fig. 6.5 shows
our predictions for Type Ibc detections of SKA per year per deg2 at 1.4 GHz with a survey
sensitivity of 50 nJy. The red curve shows the radio Type Ibc detections, assuming that
Type Ibc represents 25% of core-collapse events (Li et al., 2011b), and fradio,Ibc = 12% with
luminosity ∼ 1027 erg s−1 Hz−1 10 (Berger et al., 2003). The blue curve shows the possi-
ble detections of the sub-class of Type Ibc supernovae that display extreme radio emission
10Here we simply assume a Gaussian distribution for the luminosity function centered at the specifiedluminosity with σ = 1.
129
and hence might be powered by central engines and related to GRBs. We assume that
0.5% of all Type Ibc supernovae are powered by central engines and have luminosities of
∼ 1029 erg s−1 Hz−1 (Berger et al., 2003). We adopted the spectrum of SN 1998bw, which
is a Type Ic supernova (Weiler et al., 2002). Under these assumptions the SKA will be able
to make unbiased, untargeted detections of ∼ 130 radio Type Ibc supernovae per year per
deg2, and ∼ 20 Type Ibc supernovae that might be connected to GRBs.
Figure 6.5: Predicted detection rate of Type Ibc supernovae as a function of redshift. Inthis plot we assume the sensitivity for SKA is Smin = 50 nJy. The red curve shows all of theradio Ibc detections, assuming fradio,Ibc = 12% (Berger et al., 2003). The blue curve showsonly the detection rate for Radio Ibc with central engines, assuming 0.5% of all of the TypeIbc RSNe are powered by central engines.
Finally, we turn to SKA precursors. The EVLA11, a current leading-edge radio inter-
ferometer operating at centimeter wavelengths, is anticipated to reach a 1-σ rms noise of
σI ∼ 1 µJy or less in 10 hours, while SKA is expected to reach σI ∼ 50 nJy in 10 hours.
With data accumulated over repeated scans spanning over 1000 hours, an rms σI ∼ 5 nJy
may be reached. In synoptic surveys, we would expect EVLA to detect core-collapse events
11http://www.aoc.nrao.edu/evla
130
at a rate ∼ 160 RSNe yr−1 deg−2 over a redshift range z = 0.5 to 3 (Fig. 6.2). A sample of
this size over this redshift range will already mark a major advance in the study of cosmic
RSNe, and further motivate the full SKA. ASKAP and MeerKAT are expected to have sen-
sitivities comparable to that of EVLA (Johnston et al., 2009; de Blok et al., 2010), hence we
would expect these to detect RSNe with similar rates and redshift reach.
6.4.2 Type Ia Supernovae
If all Type Ia RSNe are dimmer than the weakest limit presented in §6.2.2, the expected SKA
detection rate is essentially zero. For example, if a typical Type Ia has a radio luminosity
equal to the lowest published limit, L = 8.1 × 1024 erg s−1 Hz−1, this can be seen with a
sensitivity Smin = 50 nJy out to a luminosity distance ∼ 300 Mpc (z ∼ 0.08). While ∼ 3900
cosmic Ia events should occur per year out to this distance over the entire sky, ≪ 1 events
are expected in the SKA field of view. More optimistically, imagine a typical Type Ia radio
luminosity is L = 1026 erg s−1 Hz−1, which is below L = 4.2× 1026 erg s−1 Hz−1, the highest
published limit (Panagia et al., 2006); here the luminosity distance increases to ∼ 1400 Mpc
(z ∼ 0.28). In this case, we find an SKA Type Ia detection rate ∼ 0.5 yr−1 deg−2, based
on the local cosmic Type Ia rate derived from SDSS-II optical data (Dilday et al., 2010),
Smin = 50 nJy, and fradio = 10%.12 We see that even optimistically, we expect fewer than
one event per SKA field-of-view per year. Even with fradio = 100%, the detection rate is still
only ∼ 5 yr−1 deg−2. Therefore we conclude that SKA will make few, if any, blind detections
of Type Ia supernovae.
Targeted radio observations to follow up from nearby optical detections will probably
be the best way to search for such events. For example, we expect 10 Type Ia events/year
in the LSST sky within ∼ 60 Mpc (z ∼ 0.015). Type Ia (or core collapse!) events within
this distance observed with Smin = 50 nJy, would be detectable at luminosities L >∼ 3.0 ×
1023 erg s−1 Hz−1. Amusingly, this is close to the radio luminosity of SN 1987A.
12This also is implied by Fig. 6.4, which is for core-collapse events that have a higher cosmic rate.
131
6.5 Radio Survey Recommendations
A key requirement for detecting weak radio emission from CSM-supernovae interactions is
improved radio interferometer sensitivity. High angular resolution – below an arcsecond
at 1.4 GHz, (Weiler et al., 2004) – is also required to avoid natural confusion and to help
identify supernovae against background galaxies. This is similar to the maximum EVLA
angular resolution at 1.4 GHz. For comparison, the maximum anticipated SKA baseline
length of 3000 km, producing angular resolution of 0.014 arcsecond at 1.4 GHz, is sufficient
to distinguish different galaxies and also to resolve galaxies as extended sources within the
observable universe with rms confusion limit of < 3 nJy at 1.4 GHz (Carilli & Rawlings,
2004).
A key science goal of the SKA is to detect transient radio sources, both known (e.g.
pulsars, GRBs), and as-yet unknown. This requires sophisticated transient detection and
classification algorithms very likely running commensally with other large surveys planned
by the SKA, such as the HI spectroscopic survey and deep continuum fields. We assume
here that SKA transient detection algorithms will encompass automated detection of RSNe.
For example, current parameterized models (Weiler et al., 1986, 1990; Montes et al., 1997;
Chevalier, 1982a,b) based on available data predict patterns of spectral index evolution char-
acteristic of supernovae in general, and supernova sub-types in particular. This information
could be exploited for RSN detection, even potentially against a background of unrelated
source variability. Broad frequency coverage is important in this regard (Weiler et al., 2004).
The SKA intrinsically is a high dynamic-range instrument, given the sensitivity implied
by the large collecting area. The most demanding SKA science applications will require a
dynamic range of 107:1. The detection of faint RSNe will require a dynamic range that falls
within that envelope.
Although the lightcurves of RSNe show great diversity, the luminosities of core-collapse
supernovae usually change much slower in radio than in optical. RSN lightcurves typically
evolve on timescales of weeks to years; a useful lightcurve compilation appears in Stockdale
132
et al. (2007). Thus the minimum survey cadence (revisit periodicity) need not be any more
frequent than this. Also, we have shown that core-collapse RSNe can be found out to high
redshift with surveys pushing down to Smin = 50 nJy. For SKA this corresponds to about
100 hours of exposure, in line with planned deep field exposures which are part of the key
science. Thus, SKA as currently envisioned is well-suited to core-collapse discovery.
On the other hand, SKA probably will not have sufficient survey sensitivity for a volu-
metric search for Type Ia events, based on our current knowledge of the cosmic Type Ia rate
and the upper limits in their luminosities set by the non-detection of these events. Follow-up
observations from other wavelengths will likely be the best way to search for Type Ia RSNe.
The small volume of the local universe will limit nearby untargeted SKA detections of
low-redshift core-collapse RSNe. We estimate only ∼ 2 core-collapse RSN detections per
year per square degree within redshift z ∼ 0.5 (assuming a 50 nJy sensitivity at 1.4 GHz
and fradio = 10%). Unless SKA has large sky coverage comparable to those of optical
surveys, it will be hard to get statistical information from such a small sample. Therefore,
targeted radio followup of optically-confirmed nearby supernovae will be crucial to build a
core-collapse RSN “training set” database needed for refining automatic identification and
classification techniques.
With detection methods optimized based on low-z radio data for optically-identified
events, radio surveys can then be used to independently detect core-collapse RSNe at high
redshift based only on their radio emission. As shown in Fig. 6.2, supernova searches at
high redshift (z & 1) will largely rely on radio synoptic surveys, the inverse of the strategy
proposed above for low-redshift domain. Surveys for core-collapse RSNe will likely not be
synoptic all-sky surveys due to operational limitations, but will likely proceed in a limited
set of sub-fields, visited over an hierarchical set of cadences to cover a range of time-scales for
general transient phenomena and multiple commensal science objectives. It is also important
to match core-collapse RSNe survey sky coverage and cadence to that used in complementary
optical surveys. LSST will repeatedly scan the whole sky every ∼ 3 days. Thus a cadence
133
∼ 1 week for RSNe sub-fields will be preferred for an SKA core-collapse supernova survey.
6.6 Discussion and Conclusions
SKA’s capability for unbiased synoptic searches over large fields of view will revolutionize the
discovery of radio transients in general and core-collapse RSNe in particular (Gal-Yam et al.,
2006). The unprecedented sensitivity of SKA could allow detection of core-collapse RSNe out
to a redshift z ∼ 5. These detections will be unbiased and automatic in that they can occur
anywhere in the large SKA field of view without need for targeting based on prior detection
at other wavelengths. With SKA, the core-collapse RSN inventory should increase from the
current number of several dozen to ∼ 620 yr−1 deg−2. EVLA should detect ∼ 160 yr−1 deg−2,
and other SKA precursors should reap similarly large RSN harvests. In contrast, intrinsically
dim RSNe such as Type Ia events and 1987A-like core-collapse explosions are unlikely to be
found blindly. However, the SKA (and precursor) sensitivities will offer the possibility of
detecting these events via targeted followup of discoveries by optical synoptic surveys such
as LSST.
The science payoff of large-scale RSNe searches touches many areas of astrophysics and
cosmology. We conclude with examples of possible science applications with the new era of
RSN survey. However, the true potential of untargeted radio search is very likely beyond
what we mention.
Non-prompt RSN emission requires the presence of circumstellar matter, so such sur-
veys will probe this material and the pre-supernova winds producing it. For core-collapse
supernovae, pre-supernova winds should depend on the metallicities of the progenitor stars
(Leitherer et al., 1992; Kudritzki & Puls, 2000; Vink et al., 2001; Mokiem et al., 2007), and
should be weaker in metal-poor environments with lower opacities in progenitor atmospheres.
This effect should lead to correlations between RSN luminosity and host metallicity, as well
as an evolution of the RSN luminosity function towards lower values at higher redshifts. For
134
Type Ia supernovae, the mass-loss rate from the progenitors depends on the nature of the
binary system, i.e., single or double degenerate (Nomoto et al., 1984; Iben & Tutukov, 1984;
Webbink, 1984). Radio detection of Type Ia supernovae will probe the mass and density
profile of the surrounding environment and hence be valuable for studying Type Ia physics
(Eck et al., 1995; Panagia et al., 2006; Chomiuk et al., 2011).
Large-scale synoptic RSNe surveys will complement their optical counterparts. While op-
tical surveys such as LSST will provide very large supernova statistics at z . 1, radio surveys
will be crucial for detections at higher redshifts. The nature and evolution of dust obscu-
ration presents a major challenge for optical supernova surveys and supernova cosmology.
Current studies suggest dust obscuration increases rapidly with redshift, but uncertainties
are large. Mannucci et al. (2007) estimate that optical surveys may miss ∼ 60% of core-
collapse supernovae and ∼ 35% of Type Ia supernovae at redshift z ∼ 2. RSN observations,
in contrast, are essentially unaffected by dust. Thus, high-redshift supernovae could be de-
tected at radio wavelengths but largely missed in counterpart optical searches. Comparing
supernova detections in both optical and radio will provide a new and independent way to
measure dust dependence on redshift. In particular, SKA will be a powerful tool to directly
detect supernovae in dust-obscured regions at large redshift, and therefore offer what may
be the only means to study the total supernovae rate, star-formation, and dust behavior in
these areas.
Additionally, radio surveys will reveal rare and exotic events. For example, some Type
Ibc supernovae are linked to long GRBs (Galama et al. 1998; and see reviews in Woosley &
Bloom, 2006; Gehrels et al., 2009), probably via highly relativistic jets powered by central
engines and will manifest themselves in extremely luminous radio emission (Woosley, 1993;
Iwamoto et al., 1998; Li & Chevalier, 1999; Woosley et al., 1999; Heger et al., 2003). Thus
one might expect radio surveys to preferentially detect more Type Ibc supernovae than
other supernova types. An unbiased sample of Type Ibc RSNe will provide new information
about the circumstellar environments of these explosions and thus probe the mass-loss effects
135
believed to be crucial to the Ibc pre-explosion evolutionary path (Price et al., 2002; Soderberg
et al., 2004, 2006a; Crockett et al., 2007; Wellons & Soderberg, 2011); in addition, a large
sample of Ibc RSNe will allow systematic study of the differences, if any, between those which
do an do not host GRBs (Berger et al., 2003; Soderberg et al., 2006b; Soderberg, 2007).
Furthermore, radio surveys give unique new insight into a possible class of massive star
deaths via direct collapse into black holes, with powerful neutrino bursts but no electromag-
netic emission (MacFadyen & Woosley, 1999; Fryer, 1999; MacFadyen et al., 2001; Heger
et al., 2003). These “invisible collapses” can be probed by comparing supernovae detected
electromagnetically and the diffuse background of cosmic supernova neutrinos (Lien et al.,
2010, and references therein). By revealing dust-enshrouded SNe, radio surveys will make
this comparison robust by removing the degeneracy between truly invisible events and those
which are simply optically obscured. Indeed, direct collapse events without explosions but
with relativistic jets are candidates for GRB progenitors. A comparison among RSNe, opti-
cal supernovae, GRBs, and neutrino observations will provide important clues to the physics
of visible and invisible collapses, and their relation with GRBs.
We thus believe that a synoptic survey in radio wavelengths will be crucial in many
fields of astrophysics, for it will bring the first complete and unbiased RSN sample and
systematically explore exotic radio transients. SKA will be capable of performing such an
untargeted survey with its unprecedented sensitivity. Our knowledge of supernovae will thus
be firmly extended into the radio and to high redshifts.
6.7 Acknowledgments
We thank Kurt Weiler, Christopher Stockdale, and Shunsaku Horiuchi for encouragement
and illuminating conversations. We are also grateful to Joseph Lazio for helpful comments
that have improved this paper.
136
Chapter 7
Conclusions and Future Work
7.1 Discussion and Conclusions
The Standard Model “Foregrounds” in nuclear and particle astrophysics are interesting by
themselves as sites of interesting and often extreme astrophysics. In addition, it is critical
to understand them in order to be able to disentangle them from signatures of new physics
and astrophysics. The lithium problem is a classic case in point in the early universe. The
effect of standard nuclear physics needs to be disentangled from more exotic particle physics
effects. The search for nuclear resonances that can destroy 7Li or 7Be effectively represents
an effort to do this. The production reactions are very well understood leaving no room for
any addition or modification to the production network. So the destruction reactions are
the only options. The result of the study revealed that despite the large number of apriori
possible resonance channels involving n,p, d, t, 3He, 4He bombarding and destroying 7Li
and 7Be nuclei, very few channels have rates high enough to bring the abundances of 7Li
down to observed values. After systematically scanning the space of resonance strengths
and energies, only three candidate resonances remain and are enlisted once again here in the
table 7.1.
This work along with few others (For e.g., Cyburt & Pospelov, 2012; Boyd et al., 2010a),
prompted experiments (Kirsebom & Davids, 2011; O’Malley et al., 2011; Charity et al., 2011)
to look for these resonances. These experiments have essentially ruled out the possibility
of these resonances solving the lithium problem (Coc, 2013). This strengthens the exciting
possibility of the lithium discrepancy being an indication of new physics beyond the Standard
137
Compound Nucleus, Initial Linit Lfin Eres Γtot Exit Exit ChannelJπ, Eex State Channels Width
9B, (5/2+), 16.71 MeV 7Be + d 1 0 219.9 keV unknown p+ 8Be∗
(16.63 MeV) unknown1 α + 5Li unknown
10B, 7Be + t 1 1 130.9 keV < 600 keV p+ 9Be∗
(11.81 MeV) unknown2+, 18.80 MeV 1 3He unknown
2 α unknown10C, 7Be + 3He unknown unknown unknown unknown p unknown
unknown unknown (Q = 15.003 MeV) α unknown3He (elastic) unknown
Table 7.1: This table lists surviving candidate resonances.
Model. Of course, the possibility of a refined interpretation of the observational data to
reconcile with theoretical predictions, in terms of improved models of stellar atmospheres is
also very promising.
The study of properties of cosmic gamma-ray sources has been a very interesting one.
The computation of the inverse-Compton emission from star forming galaxies shows that it is
subdominant compared to pionic emission in most of the Fermi -LAT range of energies, unless
starburst galaxies have a much higher e/p ratio compared to normal galaxies, suggested by
some groups (Lacki et al., 2011). Aside from this caveat, it appears that the gamma-ray
emission from star forming galaxies is dominated by the pionic emission. This is in the pure
luminosity evolution model where only the luminosity of the galaxies evolves with redshift
and not the number density. In the pure density evolution case, the pionic emission is nearly
4 times weaker than the pure luminosity evolution for normal galaxies (Fields et al., 2010).
The linear scaling of the inverse-Compton emission with the star formation rate, implies
that the IC result remains the same as in the pure luminosity evolution case. And hence
for normal galaxies, the pionic emission will dominate in either case. For starbursts this is
not as obvious. According to high e/p models (Lacki et al., 2011) consistent with low values
of magnetic fields (50µG instead of 300µG), the IC emission can dominate over the pionic
emission at Fermi -LAT energies.
From the overall gamma-ray intensity of star forming galaxies as a function of energy, they
are a guaranteed and potentially significant contribution. And this is a subject of ongoing
138
research. However, it leaves the possibility of other sources as blazars also contributing
competitively (Stecker & Venters, 2011), particularly at high energies. This means that it
is important to have other “tie-breakers” to answer the question of relative contribution
of these sources to the diffuse EGB. This provides strong motivation to look at photon
statistics of these sources as a possible tie-breaker. From the preliminary results, it seems
inevitable that a EGB has a uniformly, multicomponent origin. That is both blazars and
star forming galaxies contribute without either one being dominant. This includes photons
from resolved as well as unresolved sources. The star-forming galaxies being numerous but
dim produce a largely Poisson distribution of photon counts. This is of course, contaminated
by an irreducible Poisson emission from the Galaxy. Thus, the normalisation of the Poisson
component represents an upper limit to the contribution of star forming galaxies. The blazars
on the other hand being brighter but less numerous, are capable of producing high count
pixels. As a result, their photon distribution has a high count tail. This is a distinguishing
signature between star forming galaxies and blazars. Preliminary results indicate that blazars
contribute to 21% of all the photons consistent with other analyses (Malyshev & Hogg, 2011;
Abdo et al., 2010l) including the LAT team’s own analysis.
Gamma-ray and X-ray polarimetry represents the next frontier in high energy astronomy.
The objects of interest here in this research and sources of cosmic rays in general are promis-
ing targets for future generations of polarimeters. Blazars and GRBs are in particular the
most exciting ones. This is because, they are both touted as candidates for producing ultra
high energy cosmic rays. And in general, they are regarded as the most powerful particle
accelerators in the universe. The accelerator properties are intimately tied to the high energy
emission mechanisms probed by polarimetry. Despite the fact that the polarimeters are at
most in the soft gamma-ray energy range, the relations between different energy bands from
energy exchanges through various radiative processes obeying energy conservation (Becker,
2008), implies that polarimetry can potentially constrain cosmic ray related emission mech-
anisms. Polarised emission from GRBs has been measured by SPI on INTEGRAL (For eg
139
Gotz et al., 2009), albeit for very few sources. Blazars on the other hand may require much
more sensitive space based polarimeters to be detected in this domain. Furthermore, po-
larised emission should largely trace the leptonic emission as inverse-Compton scattering of
low energy photons by electrons and synchrotron emission are likely to be the most impor-
tant mechanisms producing polarised X-rays and soft gamma-rays. And both synchrotron
and IC with different degrees of polarisation, can be distinguished by high energy polarisa-
tion. Hadronic emission on the other hand, dont lead to polarised emission for the processes
involving primary particles or reactions. The secondary electrons produced from hadronic
processes are however capable of producing polarised emission like for the primary leptons,
however with a different spectrum. Through the secondary leptonic emission polarised emis-
sion can indirectly probe hadronic processes. It could also therefore be potentially related
to neutrinos.
7.2 Future work
Naturally one of the promising future directions is the science of high energy polarimetry.
Observationally, estimating what fraction of blazars and GRBs would be discovered by fu-
ture generations of polarimeters, is one obvious goal. This involves doing feasibility studies,
optimising detection strategies, and hopefully eventually proposing for actual observations.
Theoretically, studying the emission mechanisms of both blazars and GRBs is a major chal-
lenge primarily in details of the various competing models. But chosing models that can
make predictions testable by future polarisation missions would be one way to approach this
task. Incorporating the multimessenger approach by connecting to neutrinos is an exciting
avenue.
I will join the HESS and CTA collaborations. This provides an opportunity to extend
my gamma-ray experience to very high energies. Once again these are testing grounds for
some of those very objects that have been of interest to me during my thesis. Cosmic
140
accelerators like star forming galaxies, blazars, GRBs, etc. are all very interesting at TeV
energies. Properties of individual sources like spectra of galaxies both active and otherwise,
flaring of blazars, etc. are areas of exciting research with the improved sensitivity of HESS
II. Furthermore, the possibility of testing the line emission from dark matter with HESS II
is extremely exciting.
Thus, I wish to continue to explore particle astrophysics with newer tools and methods.
141
Appendix A
A.1 The Narrow Resonance Approximation
Consider a reaction A + b → C∗ → c + D, which passes through an excited state of the
compound nucleus C∗. We treat separately normal and subthreshold reactions, defined
respectively by a positive and negative sign of the resonance energy Eres = Eex −QC , where
Eex is the excitation energy of the C∗ state considered, and QC = ∆(A) + ∆(B) − ∆(C∗).
In general, the thermally averaged rate is
〈σv〉 =
∫
d3v e−µv2/2Tσv∫
d3v e−µv2/2T=
√
8
πµT−3/2
∫ ∞
0
dE E σ(E)e−E/T (A.1)
For a Breit-Wigner resonance with widths not strongly varying with energy, this becomes
〈σv〉 =4πωΓinitΓfin
(2πµT )3/2
∫ ∞
0
dEe−E/T
(E − Eres)2 + (Γtot/2)2(A.2)
Thus the thermal rate is controlled by the integral of the Lorentzian resonance profile mod-
ulated with the exponential Boltzmann factor.
The narrow resonance approximation has usually only been applied to the normal reso-
nance case, and assumes that the total resonance width is small compared to the temperature:
Γtot ≪ T .
142
A.1.1 Narrow Normal Resonances
In the normal or “superthreshold” case, the integral includes the peak of the Lorentzian
where E = Eres. The narrow condition then guarantees that over the Lorentzian width, the
Boltzmann factor does not change appreciably, and so we make the approximation
exp
(
−ET
)
≈ exp
(
−ET
)
(A.3)
where we choose the “typical” energy to be the peak of the Lorentzian, E = Eres. Then the
integral becomes
〈σv〉Γtot≪T ≈ ωΓinitΓfin
2(2πµT )3/2e−Eres/T
∫ ∞
0
dE1
(E − Eres)2 + (Γtot/2)2(A.4)
Furthermore, it is usually also implicitly assumed that the resonance energy is large compared
to the width: Eres ≫ Γtot. Then the integral gives 2π/Γtot, and the thermally averaged cross-
section under this approximation is given by Angulo et al. (1999),
〈σv〉Γtot≪T,Eres = ω Γeff
(
2π
µT
)3/2
e−Eres/T (A.5)
= 2.65 × 10−13µ−3/2 ω Γeff T−3/29 exp(−11.605 Eres/T9) cm3s−1 (A.6)
where the latter expression has T9 = T/109 K.
Note however, that eq. (A.2) is exactly integrable as it stands and does not require we
make the usual Eres ≫ Γtot approximation. Thus for the normal case we modify the usual
reaction rate and instead adopt the form
〈σv〉narrow,normal = 〈σv〉Γtot≪T,Eres f(2Eres/Γtot) (A.7)
Here we introduce a temperature-independent correction for finite Eres/Γtot (still with Eres >
143
0)
f(u) =1
2+
1
πarctan u . (A.8)
This factor spans f → 1/2 for Eres ≪ Γtot to f → 1 for Eres ≫ Γtot.
In practice, we adopt a slightly modified version of the correction factor in our plots.
Recall that in Figs. 2.2–2.9, we show results for lithium abundances in the presence of
resonant reactions with fixed input channels, but without reference to a specific final state.
Without the correction factor, the resonant reaction rate is characterized by two parameters,
Eres and Γeff . These two parameters are insufficient to specify the correction factor, which
depends on Eres/Γtot. Rather than separately introduce Γtot, we instead approximate the
correction factor as f(2Eres/Γeff). Because Γeff < Γtot and f is monotonically increasing, this
always underestimates the value of f and thus conservatively understates the importance of
the resonance we seek (but the approximation is never off by more than a factor of 2 in the
normal case).
A.1.2 Narrow Subhreshold Resonances
Still making the narrow resonance approximation Γtot ≪ T , we now turn to the subthreshold
case, in which Eres < 0. To make effect of the sign change explicit, we rewrite eq. (A.2) as
〈σv〉 =ωΓinitΓfin
2(2πµT )3/2
∫ ∞
0
dEe−E/T
(E + |Eres|)2 + (Γtot/2)2(A.9)
Now the integrand always excludes the resonant peak, and only includes the high-energy
wing. As with the normal case, the narrowness of the resonance implies that the Boltzmann
exponential does not change much where the Lorentzian has a significant contribution, and so
we again will approximate e−E/T ≈ e−E/T . Since we avoid the resonant peak, the choice of E
not as straightforward in the subthreshold case where we took E = Eres. This choice makes
no sense in the subthreshold case, because the e−Eres/T > 1 in the subthreshold case, yet
obviously kinetic energy E > 0 and thus the Boltzmann factor must always be a suppression
144
and not an enhancement!
Yet clearly |Eres| remains an important scale. Thus we put E = u|Eres|, and we have
examined results for different values of the dimensionless parameter u. We find good agree-
ment with numerical results when we adopt u ≈ 1, i.e., E = |Eres|. Thus for the subthreshold
case we adopt a reaction rate which is in closely analogous to the normal case:
〈σv〉narrow, subthreshold = ω Γeff
(
2π
µT
)3/2
e−|Eres|/Tf (−2|Eres|/Γtot) (A.10)
Similarly to the normal case, as the reaction becomes increasingly off-resonance, i.e., as
|Eres| grows, there is an exponential suppression. In addition, the correction factor has
limits f → 1/2 for |Eres| ≪ Γtot, and f → 0 as |Eres| ≫ Γtot. Finally, note that, as a function
of Eres, our subthreshold and normal rates match at Eres = 0, as they must physically.
145
Appendix B
B.1 The energy loss rates
The energy losses other than IC are bremsstrahlung, synchrotron and ionisation. They are
expressed as follows (Hayakawa, 1973; Ginzburg, 1979): synchrotron losses
bsync(Ee) =4
3σT c Umag
(
Ee
m c2
)2
=1
6 πσT cB2
(
Ee
m
)2
≈ 3 × 10−10 GeV sec−1
(
B
1 µG
)2 (Ee
10 TeV
)2
(B.1)
are controlled by the interstellar magnetic energy density Umag. The bremsstrahlung losses
depend on whether the medium is ionised or neutral. Due to the presence of both H I and
H II, the bremsstrahlung losses are due to both neutral and ionised hydrogen (Hayakawa,
1973; Ginzburg, 1979):
bbrem(Ee) = bbrem,ion(Ee) + bbrem,n(Ee) (B.2)
bbrem,ion(Ee) =3
πnH II α σT c Ee
[
ln
(
2 Ee
m c2
)
− 1
3
]
(B.3)
= 1.37 × 10−12 GeV sec−1( nH II
1 cm−3
)
×(
Ee
10 TeV
)[
ln
(
Ee
10 TeV
)
+ 17.2
]
(B.4)
146
bbrem,n(Ee) =3
πnH I α σT c Ee
[
ln(191) +1
18
]
(B.5)
= 7.3 × 10−12 GeV sec−1( nH I
1 cm−3
)
(
Ee
10 TeV
)
(B.6)
Here nH I and nH II are the number density of relativistic electrons in the interstellar medium,
which is equal to the number density of H I and H II.
147
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